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The Raven Paradox starts with the following statement

(1) All ravens are black.

which is equivalent to the following statement

(2) Everything that is not black is not a raven.

In all the circumstances where statement (2) is true, (1) is also true. And, if (2) is false, i.e., if we find an evidence against it, then (1) will also be false.

Now, whenever we see a Black Raven, we see an evidence which supports the statement 'All Ravens are Black'. So, if we see more and more black Raven, then our belief gets stronger and stronger that all Ravens are black.

But since the statements (1) and (2) are equivalent, so collecting evidence supporting statement (2) is also an evidence that all Ravens are black. So, if we see , for example, a red apple, then it's an evidence supporting that 'All Ravens are Black'. It's because 'A red apple' is neither black (because it is red) nor is it a Raven (because it's an Apple. Apples can't be Ravens, can they?).

This conclusion seems paradoxical, because it implies that information has been gained about ravens by looking at an apple. Also, the evidence is completely unrelated. I attempted to explain it but I'm not completely convinced by my explanation.

How can we resolve this paradox?

EDIT: It can be used to collect evidence supporting completely false statements like: 'All dinosaurs are educated'. Because we've seen plenty of things until now which are neither educated nor they're dinosaurs.

EDIT2: I think that the paradox still remains. If we have a journey and took a look at every non-black thing in the universe and found it to be non-Raven, then from this argument it should be proved that all Ravens are black. But that's paradoxical because it would mean false statements can also be proved by taking a look at everything else.

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    $\begingroup$ The statements are not equivalent! Intuitively: stating that all ravens are black has no implication whatsoever on anything else but the ravens. The statement is: $R\Rightarrow B$, where $R$ denotes all birds that are ravens and $B$ denotes the color black. The converse is not true. If memory serves this is variously denoted as modus morons or sth. like that. $\endgroup$
    – Nox
    Mar 22, 2017 at 9:41
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    $\begingroup$ I agree with all those - just because something is black does not imply it is a raven. statement one is just saying that if we find a raven, it will definitely be black $\endgroup$
    – lioness99a
    Mar 22, 2017 at 9:42
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    $\begingroup$ Uhm, what? Correct me if I'm wrong, but I read the first as $\forall x : R(x) \Rightarrow B(x)$ and the second as $\forall x : \neg B(x) \Rightarrow \neg R(x)$. $\endgroup$
    – Stefan Perko
    Mar 22, 2017 at 9:45
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    $\begingroup$ This fits Philosophy, not Mathematics. $\endgroup$
    – Asaf Karagila ♦
    Mar 22, 2017 at 10:24
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    $\begingroup$ I'm not a great one for drawing sharp boundaries between disciplines -- but this is a straight philosophy of science question. At a first approximation, a Bayesian response resolves some of the puzzlement here. But there are extended relevant discussions in the terrific Stanford Encyclopaedia of Philosophy. $\endgroup$
    – Peter Smith
    Mar 22, 2017 at 11:32

8 Answers 8

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If you wanted to prove all ravens are black, you could try to find all the ravens in the world and check they are black. Or you could try to find all the non-black things in the world, and check that they're not ravens. I don't recommend either of these approaches, especially the second.

There are approximately 16 million ravens in the world (source: International Union for Conservation of Nature), so if you see a black raven, you've got a very tiny bit of evidence for your hypothesis: you're about $6.25\times 10^{-8}$ of the way to a full proof.

There are about 10 quintillion insects in the world (source: Smithsonian). There are a lot of things in the world that are not insects (source: I really should tidy my desk). Let's say, conservatively, that there are 20 quintillion things in total. About 79% of them aren't black (source: DuPont; they were only counting cars, but I'm going to assume that's a representative sample). So if you saw one non-black thing that wasn't a raven, that gives you a really really tiny bit of evidence for your hypothesis - you're about $2.5\times 10^{-16}$ of the way to a proof, i.e. 1 red apple $\approx$ 4 nanoRavens.

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  • $\begingroup$ Okay, so 1 red apple$\approx$ 4nanoRavens. So, what will be the total contribution of all the non-black atoms we've seen since we were born? $\endgroup$
    – Dove
    Mar 26, 2017 at 1:50
  • $\begingroup$ Look at the example of $R_A$ and $R_B$ ravens that I've given in my answer. In that case, evidence that all Ravens are black has been given by looking at brown $R_B$ Ravens. And, the evidence has 'same strength' as seeing a black raven itself. $\endgroup$
    – Dove
    Mar 26, 2017 at 1:59
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    $\begingroup$ I've never seen an atom, and I don't think they have colours. But if you want to count atoms as "things", you'd have to increase the number of non-black things in the world to about $10^{50}$, so the contribution of a single atom would be so tiny it's well below the range of SI prefixes. $\endgroup$
    – Especially Lime
    Mar 26, 2017 at 6:58
  • $\begingroup$ Then what about the example of $R_A$ and $R_B$ ravens? $\endgroup$
    – Dove
    Apr 5, 2017 at 14:38
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    $\begingroup$ "Let's say, conservatively, that there are 20 quintillion things in total. About 79% of them aren't black (source: DuPont; they were only counting cars, but I'm going to assume that's a representative sample)." is now my favourite paragraph ever. $\endgroup$
    – PseudoNeo
    Apr 13, 2017 at 20:47
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An intuitive explanation of why evidence for the second statement carries less weight is that there are far more not black things than ravens.

Suppose that you are sampling marbles from a bag. Suppose that you draw 5 and they are all black; what is the probability that all in the bag are black? You need to know how many are in the bag. Try 10, 100, 1000, etc.

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First things first. From a mathematical point of view, statements (1) and (2) are clearly equivalent and that's all there is to it.

All the rest of the "argument" is completely heuristical. When I say "heuristical" I mean the words/concepts:

  • "evidence"
  • "supports a statement"
  • "belief"
  • "collecting evidence supporting statement (2) is also evidence supporting statement (1)"

which are used terribly imprecisely and happily lead to the disguised "paradox". If you care to define mathematically the words listed above, then a mathematical discussion is possible where we may or may not reach the conclusion that there is a paradox here. Such a chain of "arguments" leads to no paradox at all, at least not a mathematical one. A good example of a true mathematical paradox is Russel's famous paradox, whose conclusion lies well within mathematical definitions, and so finally forces us to abandon the notion of a universal set of all sets being a set. This however, is nowhere close, and hardly provides any interesting insights into mathematics.

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  • $\begingroup$ The second part of the argument doesn't seem imprecise or heuristical at all. "Evidence" and "(degree of) belief" are given precise definitions in probability theory, under which it's a theorem that (under certain circumstances) observing a non-black non-raven is evidence that all ravens are black. $\endgroup$
    – Tanner Swett
    Mar 22, 2017 at 12:05
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    $\begingroup$ @TannerSwett - I completely disagree. While "evidence" and "belief" could be given a precise meaning in probability theory, that is yet to happen. Moreover, the context of the discussion here has nothing to do with probability theory. Also, the underlying logic framework is classical logic, not fuzzy logic: a statement is either true or false, not "evidently true", not "I believe it is true", and not "highly likely to be true." In short, the discussion above has nothing to do with mathematics. $\endgroup$
    – uniquesolution
    Mar 22, 2017 at 12:39
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    $\begingroup$ Okay, allow me to give those words a precise meaning in probability theory. A statement $A$ is evidence for a statement $B$ if and only if $P(B|A) > P(B)$. My degree of belief in a statement $A$ is $P(A)$. (If I remember right, this is the Bayesian interpretation of probability theory.) Add a couple more axioms and assumptions, and I can give you a mathematical proof. So it's certainly not the case that this "has nothing to do with mathematics". $\endgroup$
    – Tanner Swett
    Mar 22, 2017 at 13:12
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Here an answer a bit different. It is an "extreme" case, but it may convince you that finding something that is not a raven and it isn't black is evidence that all ravens are black:

Let's say there are only 7 things in the world. 4 of them are ravens, and 3 of them are not ravens. 2 of the non-ravens are also not black.

If you want to prove that all ravens are black you could check all of the 4 ravens, so each raven is 1/4 of the total evidence you need. Or you could also check if the 2 non-black things are not ravens, so each of the two things would be 1/2 of the total evidence you need. So checking that a non-black thing is not a raven is a stronger evidence than checking that a raven is black! After you have checked that all the non-black things are not ravens, you know that no raven will be able to be non-black anymore. So the ravens will be black.

This is not paradoxical and doesn't let you prove false things. If you want to prove that all dinosaurs are educate and you are able to prove that all the non-educate things are not dinosaur, then no non-educate thing will be left to be a dinosaur anymore! And so all the dinosaur are educate. But of course if some dinosaurs are not educate you will not be able to prove that all non-educate things are not dinosaurs.

Maybe the above sentences were a bit difficult to parse, so here some Venn diagrams to look at while reading:

enter image description here

These simple set theory considerations, together with the arguments about probability that other answers gave should give you a pretty clear overall picture.

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  • $\begingroup$ I think this really solves it. $\endgroup$
    – Dove
    Jan 21, 2018 at 7:47
  • $\begingroup$ I'm a bit late to the party here, but for posterity's sake: that's not a Venn diagram. It is, however, a perfectly valid Euler diagram. A Venn diagram, by definition, must have all possible combinations of overlap and non-ovarlap, Euler diagrams have no such requirements. $\endgroup$
    – Arthur
    Apr 27 at 8:35
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I'm inclined to be Bayesian about this, and for the basic approach, see e.g. the Wikipedia article.

But it isn't trivial to get the details right. If anyone really wants a careful account of Bayesian (yep, some maths!) treatments of the Ravens Paradox, then

How Bayesian Confirmation Theory Handles the Paradox of the Ravens

is helpful (Lemmas! Theorems!!) at bringing out some of the complications while aiming to give a "Best Buy" Bayesian response.

I'm not hereby fully endorsing that account by Branden Fitelson and James Hawthorne -- but it is salutary reading for anyone inclined to think that it is trivially easy to find a satisfactory response to the issues at stake.

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If you see a number of red objects, and none of them are ravens, you gain the "evidence" that a) there exist red objects, and b) none of them are ravens. With sufficient amount of such observations, you might conclude that it's not likely for red ravens to exist (in the group of objects your samples are drawn from). Similarly for any values of red.

Of course this is only statistical evidence at best, and a heuristic at worst (as others have noted), and it only gives you information about objects in the set your samples are drawn from.

As for dinosaurs: all non-educated objects I have seen today were not dinosaurs, which could be taken as evidence that any dinosaurs in the set of objects I am likely to meet in my daily life must be educated. On the other hand, I have not seen any dinosaurs at all today, so simultaneously the number of dinosaurs in the same set is likely to be zero.

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Let $Q$ be the proposition that all ravens are black. When we make an observation $O$ of something that isn't a black raven, we update our probability estimates (in the form of the ratio of the probabilities for and against $Q$) as

$$ \frac{P(Q \mid O)}{P(\neg Q \mid O)} = \frac{P(Q \cap O)}{P(\neg Q \cap O)} = \frac{P(O \mid Q)}{P(O \mid \neg Q)} \frac{P(Q)}{P(\neg Q)} $$

So, what are the probabilities:

  • That you would observe a red apple, assuming all ravens are black?
  • That you would observe a red apple, assuming some raven isn't black?

If these are equal, then the evidence both for and against the proposition cancel out, so that observing a red apple tells us absolutely nothing about $Q$.

The paradox is the mistake of trying to make a one-sided analysis — most observations are consistent both for and against a proposition, and one must weight the evidence appropriately.

Some cases, such as seeing the sun rise being consistent with the sun not existing are very weak and for practical purposes can be ignored. But this problem is not one of those.

(yes, this is consistent; e.g. it could be a hallucination, or you're tricked into seeing a recording, or a freak atmospheric event, or ...)

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Here's how I tried to explain it but I'm not completely convinced by my explanation:

Suppose there are 60 Ravens in the world named $R_1,R_2,R_3.........R_{60}$. So, to prove all Ravens are black we start discovering all the $60$ Ravens. If we find $R_3$ to be Black, then we cross out $R_3$ from the list of 60 Ravens. So, if we cross out all the Ravens from the list by finding all of them to be of Black color, then we've proved that all Ravens are black.

Now, this happens if we try to prove all Ravens are black by collecting evidence of the second statement:

If the set of all things in the universe except Ravens is $R'$ and the set of all colors in the universe except Black is $B'$, then to prove that all Ravens are black, we've to cross out everything in the set $R'\times B'$. Both $R'$ and $B'$ are infinite sets. Now, observing a red apple is an evidence that All Ravens are black. But such an evidence doesn't contribute anything significant to our belief that All Ravens are Black. Because crossing out $(Apple, Red)$, from the infinite set $R'\times B'$ doesn't contribute anything finite to proving All Ravens are Black.

But I've assumed that there are 60 Ravens. If there are an infinite number of Ravens in the world, then observing a Black Raven also won't contribute anything finite. But I believe that observing a black Raven is quantitatively a better evidence than observing a non-black non-Raven.

What if there were two kinds of Ravens $R_A$ and $R_B$ such that all $R_A's$ were black and all $R_b$'s were brown? And both $R_A$'s and $R_b$'s are found equal in number in any area. Then, I think we could consider these two statements:

(1) All $R_A$ Ravens are black.

(2)Everything which is not black but is a Raven is not a $R_A$ Raven.

I think observing a brown $R_B$ Raven gives an evidence that all $R_A$ ravens are black. But in this case, since $R_A$ and $R_B$ Ravens are equal in number, so the evidence of seeing a brown $R_B$ Raven is of the same strength as observing a black $R_A$ Raven.

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    $\begingroup$ "Both $R'$ and $B'$ are infinite sets." Really? Forget ravens, I want to see your proof that infinite sets of concrete objects exist in our universe! $\endgroup$
    – Alex Kruckman
    Mar 22, 2017 at 13:07
  • $\begingroup$ @AlexKruckman First, colors have a continuous spectrum. So, the set of all non-black colors is definitely infinite if all of them are given a name. Now, for the set of non-Ravens: You could write the number $1$ on a wall will red paint, thereby collecting an evidence of the second statement. Then, erase it and write the rest of the natural numbers. I don't think that's finite. By the way, the eye of a Raven is a non-black non-raven, atoms are non-black non-ravens. There are infinitely many possibilities of non-black non Ravens. $\endgroup$
    – Dove
    Mar 26, 2017 at 2:02
  • $\begingroup$ Your argument about numbers shows that class of all "things" is potentially infinite, in the sense that given an infinite amount of time, I could write infinitely many different numbers on the wall. But I don't have an infinite amount of time, and it doesn't convince me that the set of all things actually in the universe is infinite. Listing two examples (an eye, an atom...) is even less convincing. I agree that there are infinitely many possibilities - this is not controversial. $\endgroup$
    – Alex Kruckman
    Mar 26, 2017 at 2:53
  • $\begingroup$ The basic idea of your answer is reasonable, though, if you replace "infinite" by "astronomically large". It's clear that the number of things in the universe is far larger than the number of ravens. $\endgroup$
    – Alex Kruckman
    Mar 26, 2017 at 2:56
  • $\begingroup$ @AlexKruckman I can't think of an explanation of my example of $R_A$ and $R_B$ ravens. In that case, you don't have to look everything else except $R_A$ ravens, but you just have to look every Raven except $R_A$ ravens. And, since $R_A$ and $R_B$ ravens are equal in number, so both evidences are of equal strength. That's why I think claiming that non-black non-ravens are a weak evidence does not resolve the paradox. $\endgroup$
    – Dove
    Mar 26, 2017 at 3:11

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