Freethought & Rationalism Archive

K · 10-18-2002, 05:30 PM

Philip:

I'm saying that your modal argument falls apart because you originally said that God has the ability to do evil because there are possible worlds where He does evil. But if you take away His ability to do evil in those worlds by making the probability zero, there is no possible world left where he does evil - hence, He doesn't have the ability to do evil in the modal sense you originally proposed.

What definition of possible are you using if you aren't saying that a possible event has a probability of occurrance which is greater than zero? How are you saying that events with zero probability of happening can happen? I'm not saying this hasn't been proposed before, but I've never heard of an event with zero probability of occurance still happening. To me that makes no sense at all. Do you have any references that I could look at?

Philip Osborne · 10-19-2002, 03:01 PM

But if you take away His ability to do evil in those worlds by making the probability zero, there is no possible world left where he does evil - hence, He doesn't have the ability to do evil in the modal sense you originally proposed.

But exactly what I have been saying, and exactly what you are trying to disprove, is that making the probability zero does not take away God's ability to do evil. Hence, you cannot assume this true in attempting to construct an argument for it.

Going back to the exactly precise spinner, we could take each individual point on that spinner (each representing some unique real number from 0 to 1) and say that since for any individual point on that spinner, there are an infinite number of points not identical to that point, the probability of the spinner landing on that point is zero. However, if we spin this exactly precise spinner at all, it must land on some point, let's say, x. This is similar to the idea that if I draw a card from a 52-card deck, then I must draw some card, although the probability of that event happening is low (1/52). Since the spinner landed on x, it follows that it is possible for the spinner to land on x, even if the probability of that happening is zero.

For references, I suggest you go to the link I posted earlier on this thread. Information on "possible worlds" is widely available on the internet.

Sincerely,

Philip

Philosoft · 10-19-2002, 06:31 PM

Philip,

This may be a stupid question, but is it really the case that your spinner example obtains zero probabilities? Would not the likelihood of the spinner landing on a particular rational number out of an infinite series lead to an undefined probability?

K · 10-19-2002, 06:39 PM

Philip:

If you are positing some kind of an infinite precision spinner which must point to an individual point, then the odds of it pointing to a given point are not zero. They are just extremely close to zero.

Because your spinner can land on any individual point, it could have landed on the point before the one it did or the point after. There is no point before or point after on the real number line. For any two points on the real number line, there are an infinite number of points in between.

HRG · 10-20-2002, 12:45 AM

Quote:

Originally posted by Philip Osborne, in part:

Concerning random number generators, there is perhaps some specific mathematical definition that I have missed. So I will use the term random number generator*, which is not bound by the restriction of choosing only integers. If you are suggesting that such a generator is impossible, I would be interested to hear an argument that demonstrates this conclusion.

There are several objections to this "random real generator" - some mathematical (1-3), some physical (4-5).

1. How would it report its results in a finite time ?

2. Problems with the independence of the Continuum hypothesis. The cardinality of the continuum is to a large degree arbitrary.

3. Problems with measure theory, the mathematical foundation of probability.

4. Such a spinner would have to be infinitely heavy. Every object has an effective extent: its Compton wavelength, which is inversely proportional to its mass.

5. If the position of an object exists with infinite accuracy, then its energy has infinite expectation value. KABOOM!

As a set of measure (aka probability) zero, a single real number is "almost nothing" in the language of measure theory (where "almost everywhere" means "everywhere except of a set of measure zero). How can you land on almost nothing?

Even if it is ridiculously unlikely, there is nothing conceptually or logically possible in an absolutely precise spinner landing on 0.35.
[/quote]
The question is whether absolutely precise spinners are conceptually or logically possible.

Quote:

A zero probability simply means that for every world on which such a spinner lands on 0.35, there are an infinite number of worlds in which it does not.

Uncountably many - which is a greater problem than appears at first glance.

Quote:

Hence the probability of the world in which the spinner lands on 0.35 obtaining is 1/infinity, or zero; but 0.35 still obtains in a possible world, and so is possible.

Probabilities are supposed to be totally additive. How do you add an uncountable numbers of zeros to obtain 1 (the spinner has to point somewhere) ?

Regards,
HRG.

Philip Osborne · 10-20-2002, 04:44 PM

I have a great deal of schoolwork, so this will likely be my last post on this thread.

Because your spinner can land on any individual point, it could have landed on the point before the one it did or the point after. There is no point before or point after on the real number line. For any two points on the real number line, there are an infinite number of points in between.

The conditions of my spinner required that each point on the circle represent some unique real number between 0 and 1, and that there be an infinite number of these points. This does not require that every single real number between 0 and 1 are accounted for on the spinner, so long as my conditions are met. Even if not every real number between 0 and 1 exists on the spinner, there can still be an infinite number of such points, and the probability of the spinner landing on any one of them is zero.

1. How would it report its results in a finite time ?

2. Problems with the independence of the Continuum hypothesis. The cardinality of the continuum is to a large degree arbitrary.

3. Problems with measure theory, the mathematical foundation of probability.

4. Such a spinner would have to be infinitely heavy. Every object has an effective extent: its Compton wavelength, which is inversely proportional to its mass.

5. If the position of an object exists with infinite accuracy, then its energy has infinite expectation value. KABOOM!

I don't have much time to respond to each of your objections, so I will only say for now that it is irrelevant whether or not the spinner itself is possible. The truth of the propositions "If an absolutely precise spinner were to exist, then the probability of it landing on point x would be zero," and "If an absolutely precise spinner were to exist, it would be possible for it to land on point x" is completely seperate from the question of whether or not such a spinner is possible. I will use another example from Wes Morriston ("What's So Good About Moral Freedom?"). Consider the following pair of propositions:

1) If humans were to be created in such a way that would violate the law of conservation of mass, then they would be featherless bipeds.

2) If humans were to be created in such a way that would violate the law of conservation of mass, then they would be feathered quadrupeds.

Even if we suppose, as you seem to be supposing, that a violation of a law such as the law of conservation is impossible, it seems clear to me that (1) is obviously true while (2) is obviously false. On this analysis, it seems that subjunctive conditional propositions with impossible antecedents can be non-trivially true. So the question of whether or not my hypothetical spinner is possible seems irrelevant.

Sincerely,

Philip

[ October 20, 2002: Message edited by: Philip Osborne ]</p>

K · 10-20-2002, 06:44 PM

Philip:

Quote:

The conditions of my spinner required that each point on the circle represent some unique real number between 0 and 1, and that there be an infinite number of these points. This does not require that every single real number between 0 and 1 are accounted for on the spinner, so long as my conditions are met. Even if not every real number between 0 and 1 exists on the spinner, there can still be an infinite number of such points, and the probability of the spinner landing on any one of them is zero.

This formulation runs into the same problem. Because your spinner has an infinite number of points, there is at least one section where between any two points, there are an infinite number of points.

Philip Osborne · 10-21-2002, 11:52 AM

This formulation runs into the same problem. Because your spinner has an infinite number of points, there is at least one section where between any two points, there are an infinite number of points.

How is this true?

Even if your argument is correct, I'm not sure that it is entirely relevant to the example at hand. It seems that my ability to randomly pick any real number between 0 and 1 does not entail that I must also be able to pick exactly the point before and exactly the point after.

Sincerely,

Philip

K · 10-21-2002, 12:28 PM

Philip:

Because the spinner has to be able to land on every single point and points have zero width. How would it be possible to say that the spinner was actually on .35 when there are an infinite number of points between .35 and any other point?

St. Robert · 10-21-2002, 09:07 PM

What leads you to believe that God has free will?

He is incapable of doing evil, because He is bound by His own rules of righteousness.

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10-18-2002, 05:30 PM	#61
K Veteran Member Join Date: Aug 2002 Location: Chicago Posts: 1,485	Philip: I'm saying that your modal argument falls apart because you originally said that God has the ability to do evil because there are possible worlds where He does evil. But if you take away His ability to do evil in those worlds by making the probability zero, there is no possible world left where he does evil - hence, He doesn't have the ability to do evil in the modal sense you originally proposed. What definition of possible are you using if you aren't saying that a possible event has a probability of occurrance which is greater than zero? How are you saying that events with zero probability of happening can happen? I'm not saying this hasn't been proposed before, but I've never heard of an event with zero probability of occurance still happening. To me that makes no sense at all. Do you have any references that I could look at?

10-19-2002, 03:01 PM	#62
Philip Osborne Regular Member Join Date: Jul 2002 Location: Cherry Hill, NJ Posts: 147	But if you take away His ability to do evil in those worlds by making the probability zero, there is no possible world left where he does evil - hence, He doesn't have the ability to do evil in the modal sense you originally proposed. But exactly what I have been saying, and exactly what you are trying to disprove, is that making the probability zero does not take away God's ability to do evil. Hence, you cannot assume this true in attempting to construct an argument for it. Going back to the exactly precise spinner, we could take each individual point on that spinner (each representing some unique real number from 0 to 1) and say that since for any individual point on that spinner, there are an infinite number of points not identical to that point, the probability of the spinner landing on that point is zero. However, if we spin this exactly precise spinner at all, it must land on some point, let's say, x. This is similar to the idea that if I draw a card from a 52-card deck, then I must draw some card, although the probability of that event happening is low (1/52). Since the spinner landed on x, it follows that it is possible for the spinner to land on x, even if the probability of that happening is zero. For references, I suggest you go to the link I posted earlier on this thread. Information on "possible worlds" is widely available on the internet. Sincerely, Philip

10-19-2002, 06:31 PM	#63
Philosoft Veteran Member Join Date: Feb 2002 Location: Southeast of disorder Posts: 6,829	Philip, This may be a stupid question, but is it really the case that your spinner example obtains zero probabilities? Would not the likelihood of the spinner landing on a particular rational number out of an infinite series lead to an undefined probability?

10-19-2002, 06:39 PM	#64
K Veteran Member Join Date: Aug 2002 Location: Chicago Posts: 1,485	Philip: If you are positing some kind of an infinite precision spinner which must point to an individual point, then the odds of it pointing to a given point are not zero. They are just extremely close to zero. Because your spinner can land on any individual point, it could have landed on the point before the one it did or the point after. There is no point before or point after on the real number line. For any two points on the real number line, there are an infinite number of points in between.

10-20-2002, 04:44 PM	#66
Philip Osborne Regular Member Join Date: Jul 2002 Location: Cherry Hill, NJ Posts: 147	I have a great deal of schoolwork, so this will likely be my last post on this thread. Because your spinner can land on any individual point, it could have landed on the point before the one it did or the point after. There is no point before or point after on the real number line. For any two points on the real number line, there are an infinite number of points in between. The conditions of my spinner required that each point on the circle represent some unique real number between 0 and 1, and that there be an infinite number of these points. This does not require that every single real number between 0 and 1 are accounted for on the spinner, so long as my conditions are met. Even if not every real number between 0 and 1 exists on the spinner, there can still be an infinite number of such points, and the probability of the spinner landing on any one of them is zero. 1. How would it report its results in a finite time ? 2. Problems with the independence of the Continuum hypothesis. The cardinality of the continuum is to a large degree arbitrary. 3. Problems with measure theory, the mathematical foundation of probability. 4. Such a spinner would have to be infinitely heavy. Every object has an effective extent: its Compton wavelength, which is inversely proportional to its mass. 5. If the position of an object exists with infinite accuracy, then its energy has infinite expectation value. KABOOM! I don't have much time to respond to each of your objections, so I will only say for now that it is irrelevant whether or not the spinner itself is possible. The truth of the propositions "If an absolutely precise spinner were to exist, then the probability of it landing on point x would be zero," and "If an absolutely precise spinner were to exist, it would be possible for it to land on point x" is completely seperate from the question of whether or not such a spinner is possible. I will use another example from Wes Morriston ("What's So Good About Moral Freedom?"). Consider the following pair of propositions: 1) If humans were to be created in such a way that would violate the law of conservation of mass, then they would be featherless bipeds. 2) If humans were to be created in such a way that would violate the law of conservation of mass, then they would be feathered quadrupeds. Even if we suppose, as you seem to be supposing, that a violation of a law such as the law of conservation is impossible, it seems clear to me that (1) is obviously true while (2) is obviously false. On this analysis, it seems that subjunctive conditional propositions with impossible antecedents can be non-trivially true. So the question of whether or not my hypothetical spinner is possible seems irrelevant. Sincerely, Philip [ October 20, 2002: Message edited by: Philip Osborne ]</p>

10-21-2002, 11:52 AM	#68
Philip Osborne Regular Member Join Date: Jul 2002 Location: Cherry Hill, NJ Posts: 147	This formulation runs into the same problem. Because your spinner has an infinite number of points, there is at least one section where between any two points, there are an infinite number of points. How is this true? Even if your argument is correct, I'm not sure that it is entirely relevant to the example at hand. It seems that my ability to randomly pick any real number between 0 and 1 does not entail that I must also be able to pick exactly the point before and exactly the point after. Sincerely, Philip

10-21-2002, 12:28 PM	#69
K Veteran Member Join Date: Aug 2002 Location: Chicago Posts: 1,485	Philip: Because the spinner has to be able to land on every single point and points have zero width. How would it be possible to say that the spinner was actually on .35 when there are an infinite number of points between .35 and any other point?

10-21-2002, 09:07 PM	#70
St. Robert Regular Member Join Date: May 2002 Location: North Carolina Posts: 170	What leads you to believe that God has free will? He is incapable of doing evil, because He is bound by His own rules of righteousness.

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