Thread: Unsolved paradoxes

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?

Originally Posted by Wayfaerer

That's all fine but I can still see how evidence for one of these two completely different claims does not support the other.

Well... it's clearly not all fine if you still think that you can imagine such evidence. It would be like saying that you can imagine a situation where a person is an unmarried man but is not a bachelor. Obviously there can be no such situation simply due to our definitions of "unmarried man" and "bachelor." For the present case, could you illustrate for us how some event could be evidence for T but not S, or vice versa?

Originally Posted by Wayfaerer

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?

The proof (summarized from Carl Hempel) was laid out pretty straightforwardly in my first post--however, I don't want to defend that proof too much because I do actually think that this paradox can be resolved in a satisfying way by taking a Bayesian perspective on the problem, a perspective which I think is actually quite intuitive. It would take a fair amount of space to fully explicate the Bayesian argument (it is based on Bayes' Theorem, a result from probability theory), but I think the important points can be summed up adequately.

Recall that the Tacking Paradox depends on two assumptions about how evidence can bear on theory: (1) if T entails E, then E is evidence for T; and (2) if E is evidence for T, and T entails P, then E is evidence for P. If we accept both of these seemingly innocent assumptions, then the paradox can always be derived.

Basically, the Bayesian accepts (1) but does not fully accept (2). Instead, the Bayesian accepts that (2) is sometimes true, but denies that it must always be the case: if E is evidence for T & P, then E is allowed to provide different degrees of support for T and for P separately, says the Bayesian. Crucially, E serves as evidence for T or P only to the extent that E would be correspondingly less likely to have obtained if T or P were NOT true. With respect to P, for example, the Bayesian says that if E is equally likely to have occurred whether it is the case that T & P or that T & not-P, then E cannot be considered as evidence for or against P, whether or not it may be evidence for T. In this way the Tacking Paradox is blocked.

To make this more concrete let's return to my running example of the roommate. We had arrived at the paradoxical conclusion that, because my observation that my roommate had slept through his alarm served as evidence for the joint hypothesis that "my roommate was out drinking last night AND the moon is made of green cheese" (by [1] from above), then it must also be the case that my roommate having slept through his alarm must also be evidence for the simple hypothesis that "the moon is made of green cheese" (by [2] from above). However, the Bayesian points out that the probability that I would observe my roommate to sleep through his alarm given that "my roommate was out drinking last night AND the moon is made of green cheese" is exactly the same as the probability that I would observe my roommate to sleep through his alarm given that "my roommate was out drinking last night AND the moon is NOT made of green cheese." This is of course not true for the first part of the joint hypothesis, the part saying that my roommate was out drinking last night. In that case, the evidence is undeniably more likely to have obtained if my roommate had been out drinking last night then if he had not. So we can say that while my observation of my roommate sleeping through his alarm DOES serve as evidence that he was out drinking last night, it does NOT serve as evidence that the moon is made of green cheese.

I think you will agree that this is a pretty satisfying answer to the paradox.

The Bayesian also has a response to the Raven's Paradox, but it is somewhat less satisfying and also more difficult to explain without appealing directly to Bayes' Theorem, which I want to avoid. Roughly put, the Bayesian bites the bullet and concedes that observing any non-black thing that is a non-raven does indeed serve as some degree of evidence that all ravens are black. However, the Bayesian maintains that, for any realistic figures, such an observation provides such an infinitesimally small degree of support for the hypothesis that we can say that it has no practical bearing on the hypothesis "in the real world." So the Raven's Paradox is not blocked in the same way that we managed to block the Tacking Paradox, but its impact is lessened considerably.

Maybe The Raven paradox isn't such a paradox after all. Since everything we can see has a color, and all colors in our visual spectrum exist (lets just consider black a color for now), maybe observing a non-black thing really is ever so slightly increasing the chances of any other entity with a color being black, since there is a finite amount of entities with definite color at any given moment and a finite amount of black things that exist.

DuB, that's just mindblowing. Especially the Raven paradox.

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?

This isn't a paradox, nor even a contradiction. There is nothing to say the barber can't shave himself, just because he additionally shaves all those who doesn't. And there's even the possible solution of him not shaving/being shaved at all.

Originally Posted by khh

DuB, that's just mindblowing. Especially the Raven paradox.

Agreed.

Originally Posted by khh

This isn't a paradox, nor even a contradiction. There is nothing to say the barber can't shave himself, just because he additionally shaves all those who doesn't. And there's even the possible solution of him not shaving/being shaved at all.

Ya, I didn't do a great job explaining it, it was supposed to say that the barber only shaves those who don't shave as Xei said.

More on the topic of induction here's one from Hegel: We learn from history that we do not learn from history.

The grandfather paradox

Imagine you build a time machine. It is possible for you to travel back in time, meet your grandfather before he produces any children (i.e. your father/mother) and kill him. Thus, you would not have been born and the time machine would not have been built, a paradox

All You Zombies paradox


A baby girl is mysteriously dropped off at an orphanage in Cleveland in 1945. "Jane" grows up lonely and dejected, not knowing who her parents are, until one day in 1963 she is strangely attracted to a drifter. She falls in love with him. But just when things are finally looking up for Jane, a series of disasters strike. First, she becomes pregnant by the drifter, who then disappears. Second, during the complicated delivery, doctors find that Jane has both sets of sex organs, and to save her life, they are forced to surgically convert "her" to a "him." Finally, a mysterious stranger kidnaps her baby from the delivery room.

Reeling from these disasters, rejected by society, scorned by fate, "he" becomes a drunkard and drifter. Not only has Jane lost her parents and her lover, but he has lost his only child as well. Years later, in 1970, he stumbles into a lonely bar, called Pop's Place, and spills out his pathetic story to an elderly bartender. The sympathetic bartender offers the drifter the chance to avenge the stranger who left her pregnant and abandoned, on the condition that he join the "time travelers corps." Both of them enter a time machine, and the bartender drops off the drifter in 1963. The drifter is strangely attracted to a young orphan woman, who subsequently becomes pregnant.

The bartender then goes forward 9 months, kidnaps the baby girl from the hospital, and drops off the baby in an orphanage back in 1945. Then the bartender drops off the thoroughly confused drifter in 1985, to enlist in the time travelers corps. The drifter eventually gets his life together, becomes a respected and elderly member of the time travelers corps, and then disguises himself as a bartender and has his most difficult mission: a date with destiny, meeting a certain drifter at Pop's Place in 1970.