What is the explanation for Zeno's paradox?

[mousesgr]

In order to walk a distance of 1 meter, we have to walk half a meter first. Similarly, we have to walk 1/4 meter before we can walk that half-meter. The same principle goes on forever. Hence we can never get started in walking a distance of 1 meter.

What is the problem with the following argument?


i know the conclusion must be wrong...but are there any problem in the premises?

 

[learningphysics]

In order to walk a distance of 1 meter, we have to walk half a meter first. Similarly, we have to walk 1/4 meter before we can walk that half-meter. The same principle goes on forever. Hence we can never get started in walking a distance of 1 meter.

What is the problem with the following argument?


i know the conclusion must be wrong...but are there any problem in the premises?

Yes... the premise that "an infinite number of distances cannot be covered in a finite amount of time" is false.

Or another way of writing it... the premise that "the sum of an infinite set of numbers adds to infinity" is false.

 

[mousesgr]

Yes... the premise that "an infinite number of distances cannot be covered in a finite amount of time" is false.

Or another way of writing it... the premise that "the sum of an infinite set of numbers adds to infinity" is false.

there's false premise, that mean the argument is valid?

 

[DaveC426913]

No, the premise is false, therefore the argument is invalid.

The sum of an infinite series can be a finite value.

For example: the sum of the series (1/2 + 1/4 + 1/8 = 1/16 ...) is not infinity, it is 1.

Thus, the time it takes to travel one meter is not infinite.

 

[whozum]

In my logic class this semester, we were told that if an argument holds a false premise, the argument can not be proven invalid, so it must be valid. We used the definition 'An argument is invalid only in the instance where its premises are all true and the conclusion is false'

 

[honestrosewater]

Probably just a confusion between validity and soundness. whozum's correct about validity. An argument is sound if it is valid and all of its premises are true.

 

[whozum]

Can you give a better definition of soundness? Like you said, we were told that it is a valid argument with true premises, but for example:

A valid argument is one where any argument posed with its form with true premises can not have false conclusions. It is deductively valid in allcases. An argument with invalid form doesn't yield a definitively true conclusion, deductively invalid even if the premises are true.

Is soundness just attributed to an argument to mean 'hey thisis a nice argument', or does it have a conclusive meaning about the argument, like validity does?

 

[honestrosewater]

Is soundness just attributed to an argument to mean 'hey thisis a nice argument', or does it have a conclusive meaning about the argument, like validity does?

Well, soundness and validity are used a lot and have several meanings in logic, but in this context, an argument is sound if it is valid and its premises are true in the real world. So this distinction between soundness and validity is really only useful outside or on the skirts of logic (like epistemology, philosophy of logic), since logic proper isn't concerned with what's true in the real world. IOW, unlike validity, the soundness of an argument doesn't matter in logic.

 

[whozum]

So validity is applies to argument forms, and soundness is applied to individual arguments?

 

[whozum]

Okay, that is actually how I see it, I've only taken a semester of logic so I'm not indepth as to the actual names of each instance. Is there a more proper/better way to looka t it?

 

[honestrosewater]

Okay, that is actually how I see it, I've only taken a semester of logic so I'm not indepth as to the actual names of each instance. Is there a more proper/better way to looka t it?

Heh, I was taking forever to figure out how to correct something, so I just deleted it. I'll put it back:

I guess you could look at it that way. For example,
If the moon is made of green cheese, then pigs can fly;
The moon is made of green cheese;
Therefore, pigs can fly.
To check for validity, you ignore the meaning of the premises and only look at the relationships between them; So you ask: If the premises were true, what would that imply about the conclusion? You'll recognize this as an instance of Modus Ponens, so it's valid. But to check for soundness, you need to look at the meaning of the premises and determine whether or not they actually are true; So you ask: Is it true that if the moon is made of green cheese, then pigs can fly? Is it true that the moon is made of green cheese? The moon isn't made of green cheese, so this argument is unsound (but still valid).

What I was trying to correct is that when checking validity you don't actually ignore the "meaning" completely, and you aren't really only looking at the relationships between premises and conclusion. An argument can be valid for three reasons:
1) its premises are inconsistent (cannot all be true together),
2) its premises logically imply its conclusion, or
3) its conclusion is a tautology (always true).
(2) is where you can just look at relationships. (1) and (3) are special cases of (2). An inconsistent set of premises logically implies any conclusion, and a tautology is logically implied by any set of premises (including the empty set of premises). In (1) and (3), the relationships don't matter. So you really need to consider all 3 cases. I don't know if that makes sense to you- I couldn't find a better way to explain it. But that is a proper way of looking at it. As an example, let P denote "Pigs can fly" and H denote "'Hurkyl' is a girl's name". Consider the arguments:
1) P, ~P therefore H.
2) (P -> H), P therefore H
3) H therefore (P v ~P)
$$\begin{array}{|c|c|c|c|c|}\hline P&\neg P&H&P \rightarrow H&P \vee \neg P \\ \hline T&F&T&T&T\\ \hline T&F&F&F&T\\ \hline F&T&T&T&T\\ \hline F&T&F&T&T\\ \hline \end{array}$$
Find the rows where all premises are true. If the conclusion is true in every row where all premises are true, then the argument is valid.
1) There are no rows where both P and ~P are true (i.e. the premises are inconsistent), so this argument is valid.
2) The premises are both true in the first row. The conclusion is also true in the first row, so this argument is valid.
3) The premise is true in the first and third rows. The conclusion is also true in the first and third rows (it's a tautology- true in every row!), so this argument is valid.
Notice that in (1) and (3), it's not the relationship between premises and conclusion that determine the argument's validity; It's the meaning of the premises and conclusions themselves.
Now to test for soundness, you just ask if the argument is valid and all premises are true in the real world.
1) Valid, but "Pigs can fly" is false, so this argument is unsound. An argument with inconsistent premises is always valid and (it's pretty safe to say) unsound. It doesn't tell you anything about the relationship between premises and conclusion.
2) Valid, but "Pigs can fly" is false, so this argument is unsound.
3) Valid, and "'Hurkyl' is a girl's name" is true, so this argument is sound. Notice that if "'Hurkyl' is a girl's name" were false, the argument would still be valid but would be unsound. An argument with a tautological conclusion is always valid, and its soundess depends solely on the truth of the premises; It doesn't tell you anything about the relationship between premises and conclusion.
So arguments of type (2) are the only interesting or persuasive ones, as far as truth about the real world goes.
Maybe this is more than you care to know; I just didn't want to leave you with the wrong information.

 

[whozum]

Thats awesome, I really appreciate that.

 

[what]

I believe it' s zeno's paradox. And he had many others as well, he liked to bother greek philosphers.

 

[mousesgr]

i want to know What is the problem with the following argument?
can this argument explained by Informal Fallacies ??

 

[honestrosewater]

You can check out this site for thorough explanations.
Which argument do you want to analyze?

 

[BenGoodchild]

Hey, just as a bit of posterity, this problem is what confused the greeks!

It isn't too hard once you've seen the trick!

Regards,

Ben

 

[The Rev]

Yes... the premise that "an infinite number of distances cannot be covered in a finite amount of time" is false.

Or another way of writing it... the premise that "the sum of an infinite set of numbers adds to infinity" is false.

Skaal.

$$\phi$$

The Rev

 

[Q_Goest]

There are an infinite number of infinitely small points which make up a distance. A point has zero dimensions, so by adding up any number of these zero dimension points, we can never come up with a distance. In order to create a distance, we need to integrate. In other words, we need to add a dimension. When going from zero dimensions to a length, we need to add a dimension, the dimension of length. Another way of looking at it is that we need to recognize the need for an additional dimension. Points have none, a line has one.

Zeno's paradox uses finite lengths, but the lengths nevertheless approach an infinitely small length as we add the sums (ie: 1/2 + 1/4 + 1/8 + ...) The result is still an infinite number of lengths which can not add up to a given value because the sum of those lengths is insufficient to give us the total length. However, if we integrated the infinite number of lengths, we could come up with the distance.

I've not seen the explanation provided in the first paragraph, but curious to know if it is used to help explain Zeno's paradox.