Paradoxes

[Scatty]

That's Prior's approach. And some people, like Chaos over there, think it is inappropriate.

[Pudpop]

Wow, and here I thought I was smart?

I read that wiki thing someone posted, and now my brain hurts...

But In my defence I'm only 12....

 

But I thought I'd take a crack at it.

 

The below statement is true.

The above statement is false.

 

/thread

 

Then realised, hey, what the hell, every paradox will end up with you saying A = "A = false", or something like that.

 

I still like these though.

[Raylen]

Suppose there is a town with just one male barber; and that every man in the town keeps himself clean-shaven:

 

Let's analyze this:

 

Lets say there exists some barber x. Alright' date=' for everyone, let's say y, they live in the town, they are clean shaven. We also know the barber is in the town.

 

1) [∀y'] ( y lives in town -> y is clean shaven)

2) [∃x] ( x is a barber & x lives in the town)

 

(The backwards A means for all, -> means if...then, backwards E is there exists, and & means and)

 

some by shaving themselves, some by attending the barber.

 

So that means we can conclude:

 

3) [∀y] ( y lives in town -> (y shaves y V -y shaves y))

Note: V = or.

 

From 1 and 2, we can substitute the barber for y, as he lives in the town.

 

So now 4) The Barber shaves himself V -(The Barber shaves himself) -> The Barber goes to the barber.

 

It seems reasonable to imagine that the barber obeys the following rule: He shaves all and only those men in town who do not shave themselves.

 

Now, we can introduce 5:

 

5) [∀x] ( Barber Shaves x <-> -(x shave x))

This means for all x, the barber shaves x if and only if they don't shave themselves (x).

 

Of course, the barber is affected by this.

 

So. 6) Barber shaves himself <-> -(barber doesn't shave himself)

 

So now. Any biconditional p<->q is truth equivalent to p&q V -p&-q, but here, q is equal to -p. So, substituting that in, we get p&-p V -p & p. Which degenerates to p&-p.

 

Now Crab Helmet, is that convincing enough?

[supersonic4ever]

I've always been confused as to what a paradox really is...

[Scatty]

I prefer more complicated ones.

 

You know the Static paradox?

 

To get from point A to point B, you have to travel from point A to point B, right? And while doing that, you will be passing through every point from A to B, right? So, you will be forced to travel half the distance to later travel it all. But after you travel half the distance, you will then have to travel one fourth of the distance, one eight, one sixteenth...and so on and so forth, so you will never reach the destination. But The distance A-B could be as small as we like it to be.

 

If so, how can we even move?

 

Any ideas? (and no, you cannot use Plank's theorems, Lambda and the expansion of the universe. that would be cheating)

[Raylen]

Also known as Zeno's paradox.

 

Again, a more sophisticated version of p & -p.

[Scatty]

Zeno's paradox? Ow, yeah,I remembered. But wasn't his with Ahile and a tortoise racing? Anyway, give an answer

[Pudpop]

In the paradox of Achilles and the Tortoise, Achilles is in a footrace with the tortoise. Achilles allows the tortoise a head start of 100 metres. If we suppose that each racer starts running at some constant speed (one very fast and one very slow), then after some finite time, Achilles will have run 100 metres, bringing him to the tortoise's starting point. During this time, the tortoise has run a much shorter distance, say, 10 metres. It will then take Achilles some further time to run that distance, by which time the tortoise will have advanced farther; and then more time still to reach this third point, while the tortoise moves ahead. Thus, whenever Achilles reaches somewhere the tortoise has been, he still has farther to go. Therefore, because there are an infinite number of points Achilles must reach where the tortoise has already been, he can never overtake the tortoise. Of course, simple experience tells us that Achilles will be able to overtake the tortoise, which is why this is a paradox.

Copied and Pasted from Wikipedia.

 

And yet, I'm confused.

[Raylen]

Logically the same.

 

Achilles wants to get to the distance between him and the tortoise, say 10 meters. He catches to the tortoise was originally, but the tortoise moves half the original distance (5m) while Achilles was catching up. Achilles catches up to the tortoise, but then the tortoise moves half the distance from before (2.5m), etc., etc., etc.

 

You can see that it's logically the same.

 

I'm surprised Crab Helmet hasn't commented.

[Scatty]

Doesn't matter, just give a damn answer.

 

As for me, I CANNOT answer this Paradox without using Plank's theorems, the expansion of the universe and Lambda.

[Raylen]

There is no answer.

 

The proof is simple:

 

The logic structure of your 'Static Paradox' is "recursive". You have p and -p. Where p is you can go to the halfway distance between A and B, but the statement itself implies -p.

 

So you have p-> -p. (If p, then not p).

 

Note the p-> q is the logical equivalent of -pVq which means -(p&-q). But here q is not p. So we have -(p& -(-p)) which is -p. So we all don't move. But we know p is true, resulting in a paradox of p&-p.

 

There is a book called Deductive Logic by Warren Goldfarb if you want to read it.

[Scatty]

You can't exactly do that here. You just meaninglessly torn the problem apart, ignoring it's actual meaning, something known as synthetic thinking. From a synthetic point of view, you are right, but from an analytical point of view, you are wrong. You know why? Because it is universally admitted that questions DO NOT HAVE a truth value.

 

And the most important part of this paradox is that it is led and finished by a question, the question being "If so, how can we even move?".

[Raylen]

Think what you like.

 

Zeno's paradoxes are a set of problems generally thought to have been devised by Zeno of Elea to support Parmenides's doctrine that "all is one" and that, contrary to the evidence of our senses, the belief in plurality and change is mistaken, and in particular that motion is nothing but an illusion.

 

Read the bolded. I'm trying to figure out the truth value for that statement. Which is your paradox, no?

[Scatty]

Maybe, but my paradox ends in a question. So, you cannot simply describe it as being "false", because questions cannot be false.

[CrabHelmet]

Now Crab Helmet' date=' is that convincing enough?

[/quote']

 

Your analysis appears to be accurate. It's essentially the same logic described earlier, except put in more formal notation. Obviously, to obtain a contradiction, we must have p & -p, since all contradictions can be reduced to that form; it is similar to the notion in mathematics based on fields that every contradiction is equivalent to, and can be reduced to, the original contradiction 0=1.

 

I don't quite follow what you're saying for Zeno's Paradox, though. Zeno's Paradox has no contradiction because it is not, in fact, a paradox, and has no real contradiction; rather, it consists of the use of wordplay and trickery to make a paradox appear where no paradox actually exists. (Specifically, it does so by claiming that an infinite series of points will take infinite time to traverse, ignoring the fact that, as the number of these points approaches infinity, the distance between them and thus the time needed to traverse them approaches zero - it's more closely related to limits and integral and differential calculus than to formal logic. It is no more a paradox than it is a paradox to say that an infinite number of points are contained in a line segment of finite length, or that an infinite number of real numbers lie between 0 and 1.)

 

Also, the reason I hadn't commented is that I'm not online twenty-four hours per day. >_>

[Dark]

To get from point A to point B, you have to travel from point A to point B, right? And while doing that, you will be passing through every point from A to B, right? So, you will be forced to travel half the distance to later travel it all. But after you travel half the distance, you will then have to travel one fourth of the distance, one eight, one sixteenth...and so on and so forth, so you will never reach the destination. But The distance A-B could be as small as we like it to be.

 

In time, the distance between point A and B will be 0.000...1, where the 0's repeat infinitely. Isn't that equal to 0?

[CrabHelmet]

In time' date=' the distance between point A and B will be 0.000...1, where the 0's repeat infinitely. Isn't that equal to 0?

[/quote']

 

Attempting to place a 1 at some "omega-plus-one" decimal place is, again, an absurdly poor definition. Approach it using the mathematical notion of limits: as the number of points becomes arbitrarily large, the distance becomes arbitrarily small (or, more simply, as the number of points approaches infinity, the distance approaches zero). Attempting to have an infinite series of zeroes followed by a 1 at the end (the end of infinity?) just leads to nonsense. (In terms of the concept of why motion works, you're still essentially correct.)

[Dark]

Bah! Didn't think of that. D;

 

And how many points are there in a line segment. INFINITY. :O

[Raylen]

Also' date=' the reason I hadn't commented is that I'm not online twenty-four hours per day. >_>

[/quote']

 

You just seem willing to do that. I mean you're title = YOUR INTELLIGENCE ALSO SURPRISING ME, and the fact the members tell me you like ripping stuff up with sarcasm.

 

As for Zeno's Paradox, I'm just analyzing the logical structure. Some event p, (the ability to move), implies -p. Zeno's Paradox's structure is simply p-> -p.

 

Any conditional p->q is only false when p is true and q is false. So, simply put, if p was false, the p-> q is true, or if q was true. So p->q = -p or q.

 

Substituting -p (as it's our consequent for q) yields: -pV-p. Which is -p. So, Zeno's Paradox simply isn't a paradox. It is simply a proof based on loose axioms that motion does not exist.

 

The logical structure is -p, and not p&-p, as it would be in all paradoxes (I'm using Prior's approach).