A Logic Paradox

[Stealers Wheel]

Someone mentioned in another thread "logic" and how all things can be measured through logic.

Here is some logic:

1. Between any two points (A,C), there is a third point exactly halfway between the other two points (B).

2. Between point (B) and point (C) there is yet another point (D) exactly halfway between point (B) and point (C).

3. In fact, there is an infinite number of points exactly halfway between any other two points, based on 1 and 2 above.

4. In order to travel from one point to any other point, one must pass through an infinite number of points that are increasingly "exactly halfway" from where one is and where one is going.

How long will that take?

In other words, can one complete an infinite number of tasks in a finite amount of time?

Put another way:

I hold in my hand a basketball. I drop the ball. It hits the floor and bounces back up. We all see it. Yet, logic says it cannot happen.

It cannot happen because before the basketball falls to the floor, it must fall to a point exactly halfway to the floor. Once there, it must then fall to a point exactly halfway between the halfway point and the floor. This process must repeat itself an infinite number of times before the ball hits the floor. Can the basketball complete an infinite number of tasks in a finite amount of time?

Logic dictates the ball cannot hit the floor.

If you doubt the logic of #4 above, here is the test: Give me any two points, tell me how far apart they are. I will divide that distance by 2 and show you the halfway point. Do it again. I will always show you the halfway point by dividing the distance by two. infinity.

Logic Paradox

 

[Excon]

Logic dictates the ball cannot hit the floor.

Wrong.

 

[Moonglow]

Someone mentioned in another thread "logic" and how all things can be measured through logic.

Here is some logic:

1. Between any two points (A,C), there is a third point exactly halfway between the other two points (B).

2. Between point (B) and point (C) there is yet another point (D) exactly halfway between point (B) and point (C).

3. In fact, there is an infinite number of points exactly halfway between any other two points, based on 1 and 2 above.

4. In order to travel from one point to any other point, one must pass through an infinite number of points that are increasingly "exactly halfway" from where one is and where one is going.

How long will that take?

In other words, can one complete an infinite number of tasks in a finite amount of time?

Put another way:

I hold in my hand a basketball. I drop the ball. It hits the floor and bounces back up. We all see it. Yet, logic says it cannot happen.

It cannot happen because before the basketball falls to the floor, it must fall to a point exactly halfway to the floor. Once there, it must then fall to a point exactly halfway between the halfway point and the floor. This process must repeat itself an infinite number of times before the ball hits the floor. Can the basketball complete an infinite number of tasks in a finite amount of time?

Logic dictates the ball cannot hit the floor.

If you doubt the logic of #4 above, here is the test: Give me any two points, tell me how far apart they are. I will divide that distance by 2 and show you the halfway point. Do it again. I will always show you the halfway point by dividing the distance by two. infinity.

Logic Paradox

What I want to know is who taught the basketball to complete the tasks...

 

[soylentgreen]

Someone mentioned in another thread "logic" and how all things can be measured through logic.

Here is some logic:

1. Between any two points (A,C), there is a third point exactly halfway between the other two points (B).

2. Between point (B) and point (C) there is yet another point (D) exactly halfway between point (B) and point (C).

3. In fact, there is an infinite number of points exactly halfway between any other two points, based on 1 and 2 above.

4. In order to travel from one point to any other point, one must pass through an infinite number of points that are increasingly "exactly halfway" from where one is and where one is going.

How long will that take?

In other words, can one complete an infinite number of tasks in a finite amount of time?

Put another way:

I hold in my hand a basketball. I drop the ball. It hits the floor and bounces back up. We all see it. Yet, logic says it cannot happen.

It cannot happen because before the basketball falls to the floor, it must fall to a point exactly halfway to the floor. Once there, it must then fall to a point exactly halfway between the halfway point and the floor. This process must repeat itself an infinite number of times before the ball hits the floor. Can the basketball complete an infinite number of tasks in a finite amount of time?

Logic dictates the ball cannot hit the floor.

If you doubt the logic of #4 above, here is the test: Give me any two points, tell me how far apart they are. I will divide that distance by 2 and show you the halfway point. Do it again. I will always show you the halfway point by dividing the distance by two. infinity.

Logic Paradox

A paradox happens only when the way we think about something does not match another way we think about something. When a paradox is created it means there is something wrong in the way we are thinking about something.

So here is an old joke to demonstrate such.

A mathematician, a physicist and an engineer were asked to answer the following question. A group of boys are lined up on one wall of a dance hall, and an equal number of girls are lined up on the opposite wall. Both groups are then instructed to advance toward each other by one quarter the distance separating them every ten seconds (i.e., if they are distance d apart at time 0, they are d/2 at t=10, d/4 at t=20, d/8 at t=30, and so on.) When do they meet at the center of the dance hall? The mathematician said they would never actually meet because the series is infinite. The physicist said they would meet when time equals infinity. The engineer said that within one minute they would be close enough for all practical purposes.

 

[soylentgreen]

Someone mentioned in another thread "logic" and how all things can be measured through logic.

Here is some logic:

1. Between any two points (A,C), there is a third point exactly halfway between the other two points (B).

2. Between point (B) and point (C) there is yet another point (D) exactly halfway between point (B) and point (C).

3. In fact, there is an infinite number of points exactly halfway between any other two points, based on 1 and 2 above.

4. In order to travel from one point to any other point, one must pass through an infinite number of points that are increasingly "exactly halfway" from where one is and where one is going.

How long will that take?

In other words, can one complete an infinite number of tasks in a finite amount of time?

Put another way:

I hold in my hand a basketball. I drop the ball. It hits the floor and bounces back up. We all see it. Yet, logic says it cannot happen.

It cannot happen because before the basketball falls to the floor, it must fall to a point exactly halfway to the floor. Once there, it must then fall to a point exactly halfway between the halfway point and the floor. This process must repeat itself an infinite number of times before the ball hits the floor. Can the basketball complete an infinite number of tasks in a finite amount of time?

Logic dictates the ball cannot hit the floor.

If you doubt the logic of #4 above, here is the test: Give me any two points, tell me how far apart they are. I will divide that distance by 2 and show you the halfway point. Do it again. I will always show you the halfway point by dividing the distance by two. infinity.

Logic Paradox

Here is another one for you.
A thought experiment.
Take an apple, cut it in half, cut that into quarters and that into eights , sixteenths, thirty two, sixty four etc etc. You are left with two choices. Either the splitting of pieces goes on for infinity or at some point there is nothing left to divide.

Take either of these answers and now reverse the process. If you chose an infinite amount of pieces then how is it possible to fit an infinite amount into a finite object of an apple. Or if you chose that at some point the pieces become nothing and cannot be divided anymore then how is it possible to reverse the process when you have nothing to start with. You cannot make something from nothing.

 

[soylentgreen]

PS> Google Georg Cantor he solved zenos paradox which is what you gave an example of.

 

[OrphanSlug]

Logic dictates the ball cannot hit the floor... Logic Paradox

Your conclusions need some refinement but ultimately your OP is not a logical paradox at all.

To qualify something as a "logical paradox" you need to have a chain of events, reasoning, methods, or decisions that lead you to a puzzling and/or counter-intuitive conclusion.

Your example is nothing more than a line between two points can be infinitely divided in half to the point of infinity, the only thing that changed with the basketball is the introduction of gravity and laws of motion. The fact that the basketball hits the floor and bounces back up is entirely irrelevant to the original line traveled during the drop, that original line (or even the line traveled by the first bounce up, or other lines) that can also be divided in half to the point of infinity.

More often than not logical paradoxes apply to Logic and Decisions, Mathematics, and Physics. They do not apply to mashed together principles that can be explained by themselves.

In terms of Philosophy, the area of the forums we are in, logical paradoxes are intended to be challenging responses to questions that operate outside the bounds of typical logical paradoxes from the above.

The classical philosophical logical paradox is Buridan's Bridge, used to narrate a logical flaw between for a fictitious event between Socrates and Plato.

Socrates wants to cross a river using a bridge guarded by Plato. Plato sees Socrates and says "Socrates, if in the first proposition which you utter, you speak the truth, I will permit you to cross. But surely, if you speak falsely, I shall throw you into the water."

Socrates responds with one and only one statement, "You will throw me into the water." No other qualifier, no other statement, no reference to tone or intention other than Socrates wanting to cross the bridge.

Plato is now in a logical paradox, with a decision to make that is counter-intuitive to the intent of guarding that bridge. If Plato throws Socrates into the water he would violate his promise of letting Socrates pass by telling the truth. If Plato allows Socrates to pass it would means Socrates spoke a lie in his reply about being thrown into the water violating his one and only one law to pass on that bridge.

The conclusion is the logical paradox, Socrates could be allowed to cross the bridge if and only if he could not be allowed to.

Now, Buridan came up with this nonsense to prove a point as later in philosophy the exercise allowed us to discuss more important logical paradoxes. Which range in covering everything from questioning systems of belief to questioning reasoning behind more emotional decisions.

Your effort gave us no such conclusion, nor does it qualify as a logical paradox from physics.

 

[Stealers Wheel]

A paradox happens only when the way we think about something does not match another way we think about something. When a paradox is created it means there is something wrong in the way we are thinking about something./QUOTE]

We can therefore conclude that an infinite number of tasks can indeed be accomplished in a finite amount of time. This must be true because the ball hits the floor.

 

[Stealers Wheel]

What I want to know is who taught the basketball to complete the tasks...

It was a trick potato. Thanks for your contribution.

 

[Stealers Wheel]

PS> Google Georg Cantor he solved zenos paradox which is what you gave an example of.

Thank you for the referral.

 

[HK.227]

Someone mentioned in another thread "logic" and how all things can be measured through logic.

Here is some logic:

1. Between any two points (A,C), there is a third point exactly halfway between the other two points (B).

2. Between point (B) and point (C) there is yet another point (D) exactly halfway between point (B) and point (C).

3. In fact, there is an infinite number of points exactly halfway between any other two points, based on 1 and 2 above.

4. In order to travel from one point to any other point, one must pass through an infinite number of points that are increasingly "exactly halfway" from where one is and where one is going.

How long will that take?

In other words, can one complete an infinite number of tasks in a finite amount of time?

Put another way:

I hold in my hand a basketball. I drop the ball. It hits the floor and bounces back up. We all see it. Yet, logic says it cannot happen.

It cannot happen because before the basketball falls to the floor, it must fall to a point exactly halfway to the floor. Once there, it must then fall to a point exactly halfway between the halfway point and the floor. This process must repeat itself an infinite number of times before the ball hits the floor. Can the basketball complete an infinite number of tasks in a finite amount of time?

Logic dictates the ball cannot hit the floor.

If you doubt the logic of #4 above, here is the test: Give me any two points, tell me how far apart they are. I will divide that distance by 2 and show you the halfway point. Do it again. I will always show you the halfway point by dividing the distance by two. infinity.

Logic Paradox

Late, but a few observations.

Achilleus and the turtle, as someone already mentioned, has several possible solutions.
One is to consider that you cannot really remove time from space-time, and that the question is invalid. Even if you do, the paradox can be solved by introducing a smallest possible measurement, hereby introducing a point at which distances can no longer be halved. A third possibility is to observe that a paradox has never been observed in "the wild" and that the question is therefore irrelevant, all of which have been covered to some degree in this thread.

But it is however interesting to note that paradoxes occur all the time in binary computing, binary options consisting of two, and only two, absolute logical values. This is different from nature, where the closest thing we have to an absolute value is (afaik) the Planck unit, and where what we consider paradoxes tend to resolve themselves, whereas in binary computing they will lead to errors, freezes, or even crashes. It is important to note that binary computing does not contain actual absolute binary values, the CPU unit merely interprets them as such.

This leads to the conclusion that the sum of generelizations that we consider logical may be an imperfect tool for looking at and describing the universe.
This will come as no surprise to anyone with a STEM education, as they are constantly adjusting the aforementioned sum of generelizations with new research. But for the rest of us, the CPU comparison is a useful reminder that what we interpret is not the same as actual thing in ways we are unable to perceive.

 

[Stealers Wheel]

Late, but a few observations.

Achilleus and the turtle, as someone already mentioned, has several possible solutions.
One is to consider that you cannot really remove time from space-time, and that the question is invalid. Even if you do, the paradox can be solved by introducing a smallest possible measurement, hereby introducing a point at which distances can no longer be halved. A third possibility is to observe that a paradox has never been observed in "the wild" and that the question is therefore irrelevant, all of which have been covered to some degree in this thread.

But it is however interesting to note that paradoxes occur all the time in binary computing, binary options consisting of two, and only two, absolute logical values. This is different from nature, where the closest thing we have to an absolute value is (afaik) the Planck unit, and where what we consider paradoxes tend to resolve themselves, whereas in binary computing they will lead to errors, freezes, or even crashes. It is important to note that binary computing does not contain actual absolute binary values, the CPU unit merely interprets them as such.

This leads to the conclusion that the sum of generelizations that we consider logical may be an imperfect tool for looking at and describing the universe.
This will come as no surprise to anyone with a STEM education, as they are constantly adjusting the aforementioned sum of generelizations with new research. But for the rest of us, the CPU comparison is a useful reminder that what we interpret is not the same as actual thing in ways we are unable to perceive.

So you're saying that the premise, between any two points is a third point exactly halfway between the first two, is false.

 

[HK.227]

So you're saying that the premise, between any two points is a third point exactly halfway between the first two, is false.

Not exactly. I'm saying there is no particular reason to believe that it must be true.

 

[AlphaOmega]

Excellent thread. Lets ask it a different way. I am crossing the street. In order to get to the other side I travel in increments exactly half the distance of my previous step. Do I get to the other side?

 

[Stealers Wheel]

Excellent thread. Lets ask it a different way. I am crossing the street. In order to get to the other side I travel in increments exactly half the distance of my previous step. Do I get to the other side?

Only if you are able to perform an infinite number of tasks in a finite amount of time.

 

[Moonglow]

Someone mentioned in another thread "logic" and how all things can be measured through logic.

Here is some logic:

1. Between any two points (A,C), there is a third point exactly halfway between the other two points (B).

2. Between point (B) and point (C) there is yet another point (D) exactly halfway between point (B) and point (C).

3. In fact, there is an infinite number of points exactly halfway between any other two points, based on 1 and 2 above.

4. In order to travel from one point to any other point, one must pass through an infinite number of points that are increasingly "exactly halfway" from where one is and where one is going.

How long will that take?

In other words, can one complete an infinite number of tasks in a finite amount of time?

Put another way:

I hold in my hand a basketball. I drop the ball. It hits the floor and bounces back up. We all see it. Yet, logic says it cannot happen.

It cannot happen because before the basketball falls to the floor, it must fall to a point exactly halfway to the floor. Once there, it must then fall to a point exactly halfway between the halfway point and the floor. This process must repeat itself an infinite number of times before the ball hits the floor. Can the basketball complete an infinite number of tasks in a finite amount of time?

Logic dictates the ball cannot hit the floor.

If you doubt the logic of #4 above, here is the test: Give me any two points, tell me how far apart they are. I will divide that distance by 2 and show you the halfway point. Do it again. I will always show you the halfway point by dividing the distance by two. infinity.

Logic Paradox

Achilles and the tortoise...

 

[Stealers Wheel]

Not exactly. I'm saying there is no particular reason to believe that it must be true.

So you're saying that it is possible that a given distance cannot be divided by two?

 

[stevecanuck]

Something else to consider. There are two types of infinities: infinitely large, and infinitely small. If you were to reverse the question and say you must travel twice as far each time, then yes you would never complete the task.

 

[Visbek]

Someone mentioned in another thread "logic" and how all things can be measured through logic.

Here is some logic...

That is Zeno's paradox of motion, which dates back to the 5th Century BCE.

The modern answer is based on calculus, namely that an infinite geometric series can in fact converge. As Wolfram Alpha phrases it, "the infinite number of "half-steps" needed is balanced by the increasingly short amount of time needed to traverse the distances."

There are plenty of other paradoxes, such as the Russell's Paradox:

"R is the set of all sets that do not contain themselves. Is R contained in R?"

(This type of set construction is typically ruled out by ZFC.)


Or, Quine's paradox, which avoids the self-reference underlying many paradoxes:

"yields falsehood when preceded by its quotation" yields falsehood when preceded by its quotation.


Anyway.... Some terminological clarifications are probably required. "Logic" is actually a specific field which is used to determine the validity of inferences. Modern logicians don't see their tools as "measuring everything," and have known for quite some time that there are boundaries to logic. The most critical of these was articulated by Kurt Godel, who proved that any axiomatic logical system capable of constructing arithmetic will either be inconsistent or incomplete, but cannot be both. (I.e. if your logical system is truly complete, then it will allow inconsistencies, and that renders your logical system useless; and if you exclude those inconsistencies, then it can't be complete.)

The person you were talking to may have meant terms like "reason" or "rationality," which is not the same thing. It's a common example of using a technical term according to a non-technical meaning.

 

[AlphaOmega]

Only if you are able to perform an infinite number of tasks in a finite amount of time.

Nope, you will never get to the other side even if you have an infinite amount of time. The concept is very black hole'ish

 

[HK.227]

So you're saying that it is possible that a given distance cannot be divided by two?

Not only that. I'm saying that it is quite possible that "two" does not exist.

 

[RAMOSS]

So you're saying that it is possible that a given distance cannot be divided by two?

If distance is digital rather than analog.

 

[Stealers Wheel]

If distance is digital rather than analog.

Could you post a digital distance that is not divisible by 2?

 

[RAMOSS]

Could you post a digital distance that is not divisible by 2?

The planck length. You can come up with a number, but that number does not have a meaning in when it comes to relating to the physical world.

 

[Stealers Wheel]

The planck length. You can come up with a number, but that number does not have a meaning in when it comes to relating to the physical world.

So it is not possible to calculate a half planck?

If a planck has a length, why can't that length be divided by 2?