Thread: Unsolved paradoxes

Well I don't know if this has a formal name but as a consequence of special relativity, two people witnessing the same event but traveling at different speeds will have different accounts of the same event because the way we experience time depends on how fast we are traveling.

There is also the barber paradox. If a barber shaves everyone who do not shave themselves, then who shaves the barber?

To explicate the 'different accounts' thing; for one person, two events can occur simultaneously, but for another at a different velocity, the events occur at different times. The really paradoxical thing is that they're both correct. It's not actually that hard to deduce; google for simultaneity and the train thought experiment.

Originally Posted by stormcrow

There is also the barber paradox. If a barber shaves everyone who do not shave themselves, then who shaves the barber?

Simple. Someone else.

Right. So the barber does.

Originally Posted by stormcrow

There is also the barber paradox. If a barber shaves everyone who do not shave themselves, then who shaves the barber?

That isn't a paradox, it's just a contradiction. There is no answer. Whether he shaves himself or not, the requirements aren't met.
@Darkmatters If someone else shaves him, he does not shave himself. And therefore, according to the rule, he needs to shave himself.
@Xei It's implied that he shaves people only iff no one else does.

I still think the Hangman's Paradox is unsolved.

And there are others like the heap paradox, which seem less impressive to me. But, if you look at one of them, it is kind of interesting that the argument seems undeniably sound, yet the conclusion is false:

P) If a girl is young, the next day she will still be young.
C) If this is applied every day, after 60 years she will still be young.

Originally Posted by Xei

Right. So the barber does.

Well whoever does it would then be a barber yes. I don't see any contradiction or paradox.

The Barber is a person. If somebody else shaves him then he doesn't shave himself. As he shaves everybody who doesn't shave themselves, this means he shaves himself. Which means somebody else doesn't.

Ok, I get it. Hurt my brain a little but I get it now.

So basically, like most paradoxes (paradoces?)it's just a word game that doesn't apply to reality an any way. The initial statement contains an arbitrary made-up rule that's then contradicted in the second one.

I suspect if we were able to examine them closely enough eventually we'd realize this is true for every paradox. Just word games.

This statement is false.

Whoopdidoo, logical paradoxes are boring.

Yeah, those types are boring.

One paradox I've found interesting involves the stock market. Over the years I've noticed that stock market crashes tend to occur in the September/October time period. Now, if I noticed this, then actual stock traders have certainly noticed it. But if they know it's going to happen (or if it's more likely than pure randomness would allow) then the rational choice would be to short sell your portfolio just prior to this time period (say, August). But if all the big players in the market do this, then the market would crash in August. But they also know this, so the market will actually crash in August. So they should they should really short sell their portfolios in July. But everyone knows this, so the market should crash in July. They should really be short selling in June. But they all know this... on and on.

It seems as though the rational choice is to always short sell immediately. So there are two paradoxes: why does the market go up over time if the players are rational, and why do the crashes actually occur in September/October, even though everyone knows it's going to happen?

Originally Posted by cmind

if the players are rational

solv'd

Originally Posted by Xei

solv'd

I can't see an obvious reason for irrationality though... It could only be an emergent property.

My favourite is the grandfather paradox: Suppose a man traveled back in time and killed his biological grandfather before the latter met the traveler's grandmother. As a result, one of the traveler's parents (and by extension the traveler himself) would never have been conceived. This would imply that he could not have traveled back in time after all, which means the grandfather would still be alive, and the traveler would have been conceived allowing him to travel back in time and kill his grandfather. Thus each possibility seems to imply its own negation, a type of logical paradox.

Grandfather paradox - Wikipedia, the free encyclopedia

Also there is the Monty Hall problem which is quite interesting: Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1 [but the door is not opened], and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat. He then says to you, "Do you want to pick door No. 2?" Is it to your advantage to switch your choice?

Monty Hall problem - Wikipedia, the free encyclopedia

Does a set of all sets contain itself?

If it existed then obviously it would, because it contains all sets. That is not actually paradoxical in any way.

It doesn't exist however, for completely different reasons.

Any questions about infinite sequences of sets would be easily dealt with using simple analysis techniques.

Depends what exactly you mean; Indeed's question is related to set theory and cardinality, not matters of analysis, unless there's something I don't know about.

Some of my favorite "paradoxes" have to do with induction. Below I describe the Raven's Paradox and the Tacking Paradox.

The Raven's Paradox

Let's first make two very reasonable assumptions about the concepts of evidence and theory and how they relate to one another.

First we assume that if we have a generalization T (e.g., a theory that says "all lawyers are greedy"), and we observe an instance E of this generalization (i.e., we observe a greedy lawyer), then this instance E serves as some positive evidence for T.

Second we assume that if E serves as evidence for T, and T is logically equivalent to another theory S, then E serves as evidence for S.

As I said, both of these assumptions seem entirely reasonable.

Now consider the theory T that "all ravens are black." Note that this theory is logically equivalent to the theory S that "all non-black things are non-ravens." Now imagine that I have just seen a green apple. Since this apple is a non-black thing which is also a non-raven, then by the first assumption we made above, this serves as evidence for S. But since S is logically equivalent to T, then by our second assumption, this also serves as evidence for T. So my observation of a green apple is evidence that all ravens are black. But this seems absurd.

The Tacking Paradox

We start with two more assumptions.

First we assume that if E is a logically necessary consequence of T, then E serves as evidence for T. For example, say we have a T which says "my roommate was out drinking last night." We combine this with the proposition "if T, then my roommate will sleep through his alarm clock's ringing." Now I observe the event E that my roommate has slept through his alarm. So on this assumption, E serves as evidence that my roommate was out drinking last night.

Second we assume that if E serves as evidence for T, and S is some other logically necessary consequence of T, then E serves as evidence for S. Continuing with the roommate example, suppose we add a second proposition that "if T, then my roommate will sleep with his shoes still on." So on this assumption, observing that my roommate has slept through his alarm clock's ringing serves as evidence that he has also slept with his shoes still on, by virtue of the fact that the former serves as evidence that my roommate was out drinking last night.

Again, these are reasonable assumptions.

But now let's take any other proposition P, such as "the moon is made of green cheese." Since T from above entails E, then by hypothesis, T & P (that is, my roommate was out drinking last night AND the moon is made of green cheese) also entails E (that my roommate will sleep through his alarm clock's ringing). So by our third assumption, E serves as evidence for T & P. Now, it is obvious that T & P entails P (trivially). But then, by our fourth assumption, it follows that E serves as evidence for P -- that is, my observation that my roommate has slept through his alarm clock's ringing is evidence that the moon is made of green cheese. But again, this seems absurd.

Here is a kind of boring paradox:
If I succeeded at failing which have I done? I think that you have succeeded personally but some would take this as a paradox.
Also isn't a paradox by nature unsolvable?

DuB you are basically the best poster on DV.

I think it's clear that there are some big problems with classical logic and philosophy of science. I recently read an article about metamathematics and a concept I think they called 'pseudoconsistency' about similar things; basically regarding the rift between classical logic and intuitive reasoning. The principle of explosion, which I'm sure you know about, basically relies on exactly the same trick; using formal operators (like AND) to make conclusions about things they clearly should have no relevance to.

I also think this stuff will ultimately turn out to have great relevance to neuroscience. It's something I intend to read much deeper about when I'm not so busy on my degree.

Originally Posted by DuB

Second we assume that if E serves as evidence for T, and T is logically equivalent to another theory S, then E serves as evidence for S.

How are theories T and S reasonably equivalent? T is making a statement about the color of ravens while S is making a statement about the entity of other things.

Originally Posted by DuB

Since T from above entails E, then by hypothesis, T & P (that is, my roommate was out drinking last night AND the moon is made of green cheese) also entails E

How does proposition P logically fit into the relationships of E,S, and T at all?

Originally Posted by Wayfaerer

How are theories T and S reasonably equivalent? T is making a statement about the color of ravens while S is making a statement about the entity of other things.

They are not only "reasonably" equivalent, they are exactly equivalent. We can even say that they are synonyms. I think this can be made clear by thinking about the two theories in terms of possible worlds.

First imagine a world where T is true, that is, a world where all ravens are black. There may be all kinds of other things besides ravens in this world, but in this world, if the thing in question is a raven, then it is a black thing. Call this a T-world.

Next imagine a world where S is true, that is, a world where all non-black things are non-ravens. There may be things of all colors besides black in this world, but in this world, none of these non-black things are ravens. Call this an S-world.

Now let's try to think of a way within T-world in which S could be falsified. That is, we are trying to imagine a world where T is true but S is false. But it doesn't seem that this is possible. In order to falsify S, we would have to find some non-black thing which is a raven. But in T-world, by definition, all ravens are black. And since finding a non-black thing which is a raven would entail that not all ravens are black, then it can never be possible to falsify S in any T-world.

Conversely we can try to think of a way within S-world in which T could be falsified. That is, we are trying to imagine a world where S is true but T is false. But again, this doesn't seem possible. In order to falsify T, we would have to find a raven which is non-black. But in S-world, no non-black things are ravens. Just like above, by definition we could never falsify T in any S-world.

This is of course the case because T is logically equivalent to S. Any world in which T is true is also a world in which S is true, and vice versa. No other state of affairs is logically possible.

Originally Posted by Wayfaerer

How does proposition P logically fit into the relationships of E,S, and T at all?

It doesn't have any prior relevance at all. That's the entire point of the Tacking Paradox: that we can take any arbitrary proposition P, no matter how ridiculous or how far removed from the relevant theory it may be, and yet--by "tacking" P onto T by use of a conjunction--we can rigorously prove that evidence for T must also be evidence for P. The fact that this flies completely in the face of our intuitions is why we call it a paradox. If you're asking me for a way to make it seem non-paradoxical, I'm afraid I cannot.

Originally Posted by DuB

I think this can be made clear by thinking about the two theories in terms of possible worlds.

That's all fine but I can still see how evidence for one of these two completely different claims does not support the other. Evidence to support them seeks two different types of properties for two different types of things.

Originally Posted by DuB

by "tacking" P onto T by use of a conjunction--we can rigorously prove that evidence for T must also be evidence for P. The fact that this flies completely in the face of our intuitions is why we call it a paradox. If you're asking me for a way to make it seem non-paradoxical, I'm afraid I cannot.

How can you rigorously prove this? It seems to me you just disproved this notion. Why not conclude this kind of thinking is wrong in the first place rather than seeing it as some confounding logical glitch?