How is the Raven Paradox a paradox?

In My Humble Opinion
1

So this is the logical “paradox” discussed
here (also no imbedded YouTube, booo bad Cecil :wink: )

That says that because these two statements are logically equivalent:
All Ravens are black
All non-Black objects are non-Ravens

Then these statements are too
Finding a black raven is support for the hypothesis that all Ravens are black.
Finding a non-black non-raven is support for the hypothesis that all non-black objects are non-ravens.

Which in turn means:
Finding a non-Black non-Raven is support for the hypothesis that all ravens are black

So if you find a white shoe, that i support for the hypothese all ravens are black.

But that seems trivially true to me, I don’t see the paradox, finding a non-black non-raven object does provide support for that hypothesis, albeit by a minuscule amount. The first explanation given in that videos seems obviously correct, I don’t get the paradox or the reason the other two (IMO much more tenuous) explanations in the video are needed. It’s a little counter intuitive, but that doesn’t make it a paradox.

I mean if instead of saying the statements are describing all the things in the universe, you preface this by saying we are talking about the set of model things in a small box I am holding it is trivially true. If I pull out a white shoe that supports the assertion that all the ravens in the box are in fact black (in fact if it is the last thing in the box and all the ravens pulled out so far are black, it proves the statement true). It seems just as true for a box containing all the things in the universe.

What am I missing?

2

Yes, you have it. Seeking out nonblack things and finding that they’re all nonravens does support the hypothesis that all ravens are black… it just supports it extremely weakly. And of course, there must be some explanation for why it’s not a paradox, because paradoxes, by definition, don’t exist.

Here’s another paradox (or rather, apparent paradox): Suppose I take the hypothesis “No human can ever be nine feet tall”, and I set out to test it by finding a bunch of humans, and measuring them. Every time I find a human who’s less than nine feet tall, that should increase my confidence in my hypothesis (even if only a very slight increase), right?

Well, now, in the process of sampling humans, suppose I come across Robert Wadlow, and measure him to be 8’ 11.5". Well, he’s another human who’s less than 9’ tall, after all, so his existence should make me slightly more confident that no human can ever be 9’ tall. And yet… Finding him actually makes me much less confident in that hypothesis, not more, because I say to myself that if he can exist, surely it’s possible for someone to exist a mere half-inch taller than him, right?

3

Different ways to look at this. Not disproving the statement supports it, by a trivial amount as you say. In trials lawyers may ask an expert witness if evidence is consistent with some conclusion. The evidence may be consistent with the conclusion simply by not disproving it.

4

Yeah and that shows why falsifying is much better. I mean a single black raven or non-Black non-raven, is a very small amount of support for the hypothesis. A single purple raven is 100% proof that the hypothesis is false.

But that doesn’t change the fact that either a single black raven or non-Black non-raven is support for the hypothesis.

5

I agree with you and the other responses, I don’t think you’re missing anything, provided that there is a finite number of objects (perhaps also with an infinite number, but I don’t know how to think about that).

Suppose the total number of unexamined non-black objects is N. If you examine 1 of these N objects and find that it is a raven, the hypothesis is disproven. Conversely, if you examine 1 of the non-black objects and find that is not a raven, there remain only N-1 non-black objects that could be raven, so the hypothesis is now slightly more likely to be true - by a factor of N/(N-1) if there were equal prior uncertainty about each of the N objects.

(ETA that factor isn’t quite right if you do the Bayesian math, but it’s that order of magnitude.)

6

Very, very loosely speaking it’s a variant of “that which does not kill me makes me stronger.” Every non-counterexample slightly enhances the probabilistic validity of the proposition.

The truth of course is that most of what doesn’t kill you doesn’t really make you stronger; you’re simply unaffected. And most of what darn near kills you makes you weaker, at least until you heal up. :wink:

But I hope you can see the metaphorical similarity between the raven idea and this one. Most of the non-black non-ravens you encounter don’t really make the “every raven is black” argument stronger. Except in the very weakest sense of “stronger”.

7

There are Ravens that lack melanin. Just sayin.

8

Yes, and they’re noted for their combat maneuvers which tend to support their allies.

9

Nitpick: 8’11.1". Something I remember from poring through my Guinness books when I was a kid.

To the OP, I don’t see a paradox.

10

The paradoxon, if you are searching for one, resides in the fact that a non-black/non-raven object proves anything you want to besides that ravens are black, for instance that grass is green. And that grass is red, or violett if you prefer, or that unicorns are blue.
Simply put: a non-black/non-raven object proves nothing about ravens, it just makes you believe it does, because you want to. But it is just confirmation bias.
It is not a paradox, though. It just shows your prejudices. The premise is about psychology, not logic.

11

Well, a non-black/non-raven object proves nothing beyond its own existence.

It suggests that any proposition whatsoever that is not inconsistent with the object, has not yet been disproven by this particular example.

So just keep pulling up more non-counterexamples until you’ve drained the entire Universe and then you’ll have some proof. Until then you have no proof; just a pile of suggestions. And given the size of the Universe, your pile of suggestions is necessarily going to be a very, very tiny fraction thereof. So tiny as to make ignoring that pile the smarter move as it applies to deciding the truth of your proposition.

See also “How many grains of sand constitutes a pile?”

12

One quibble is that the statements are not equivalent. There is a hidden premise at work; that ravens exist.

Consider these pair of statements:

All unicorns are pink.
All non-pink objects are non-unicorns.

Suppose you were somehow able to examine all of the non-pink objects in the universe. You would not observe any unicorns. Would you have proven that unicorns are pink?

Or consider this pair of statements:

All unicorns are purple.
All non-purple objects are non-unicorns.

You then got to the supermarket and walk through the produce department. You observe yellow bananas, red apples, and green cucumbers. All of these objects are non-pink and non-purple. So therefore they all provide equal evidence that all unicorns are pink and that all unicorns are purple (as well as equal evidence that all ravens are black and all ravens are white).

13

I think I might see why it feels like a paradox. When you think of a white shoe, you tend to think of it as a shoe with one additional trait (white). Why should learning something about a shoe tell you anything about a raven? It feels silly.

Instead, think of it as an entity with two known traits: Shoeness, and Whiteness. Imagine you learned of the whiteness first: over there is a white thing, but you’ll have to get closer to tell what it is.

At that point, given what you believe to be true, you’ll see this white thing and think, “Better not be a raven, or my world is a lie!” When you learn that, sure enough, that white thing isn’t a raven, you’ll feel vindicated.

We intuitively prioritize Shoe as the more important trait of a white shoe, and that leads to us seeing it as a trivial exercise and even a paradox.

14

I think something is wrong here.

The statements “all Ravens are black” and “all non-Black objects are non-Ravens” are equivalent.

There is a hidden hypothesis that “if [whatever] exists”, but a false hypothetical can imply anything. The statement “all unicorns are pink” is true, if there are no unicorns. And the statement “all unicorns are purple” is also true if there are no unicorns.

Finding a yellow banana is consistent with the statements “all bananas are yellow” and “all non-yellow things are not bananas”. That latter statement is consistent with both “all unicorns are pink” and “all unicorns are purple” because unicorns are not bananas and neither pink nor purple are yellow.

I think the “paradox” comes from the connotation that the terms “supports” is somehow stronger than the phrase “is consistent with”. Failing to be a counter-example is the least dispositive evidence possible.

15

I would feel that existing is a necessary prerequisite to having qualities like a color. So if a unicorn doesn’t exist, it can’t be pink or purple.

The reason I feel the two statements are not equivalent is because they provide differing evidence to two separate premises; the spoken and the unspoken.

Let’s make the implied premises explicit:

If ravens exist, then all ravens are black.
If ravens exist, then all non-black objects are non-ravens.

Now let’s gather some hypothetical evidence. We’ll got out looking for ravens and non-black objects. When we find a raven, we’ll observe what color it is. When we find a non-black object, we’ll determine its ravenosity.

We find a number of non-black objects first (no surprise). Let’s say an apple, a banana, a baseball, a cucumber, a firetruck, a moose, and a rose. We confirm that all of these objects are not black and that none of them is a raven. So all of these things add a small degree of evidence to support (but not yet prove) our premise that all non-black objects are non-ravens. But they provide no evidence to support or oppose the associated premise that ravens exist.

Then we find a raven. We verify that it is black. We have found some more evidence to support (but not prove) the premise that all ravens are black. But we have also now found evidence to prove the associated premise that ravens exist.

So the two statements are not equivalent because the evidence you find to support one addresses different premises than the evidence you find to support the other.

And, as a bonus, here’s a different way in which the two statements are not equivalent. They have different degrees of falsifiability.

Consider the existence of counter-examples. Let’s say you find a bowling ball during your search for evidence. It’s a black object. But it’s not a raven. Does it disprove the hypothesis? No, your hypothesis encompasses the possibility that there are black objects which are non-ravens. So one counter-example does not disprove the hypothesis. Even when you find multiple counter-examples of this type, they do not disprove your hypothesis.

Now suppose you find an albino raven. You’ve now found a raven and it’s not black. You now have to accept that your hypothesis that all ravens are black is not correct. This single counter-example, by itself, disproves your hypothesis.

So the statement “all ravens are black” can be disproven by a single counter-example while the statement “all non-black objects are non-ravens” cannot be disproven by a single counter-example. Therefore, the two statements are not equivalent.

16

So IIRC this is just an ambiguity introduced by translating from the formal logical equations this was originally described using, to “human readable” English.

It’s been a long time but I think logical calculus unambiguously distinguishes between “there are zero or more unicorns, and all of them are pink” vs “there are one or more ravens, and all of them are black”

17

I am afraid they are not: all ravens may be black, but not all black objects are ravens. But if all ravens are black, then all non-black objects are non-ravens, without exception. There is an asymmetry here that breaks the equivalence.

18

But the statement is not about black objects it’s about non-black objects. Black objects can be ravens or non-ravens. But all non-Black objects MUST be non-ravens if all ravens are black. They are equivalent statements

19

Yes it can. It’s disproven with exactly the same single albino raven, it’s a non-Black object that is not a non-raven.

20

A white raven is a counter example to both.

“All ravens are black” means “if a thing is a raven, then it is black”. A white raven satisfies the hypothesis, but fails the result.

“All non-black things are non-ravens” means “if a thing is not black, then it is not a raven”. A white raven satisfies the hypothesis, but fails the result.