All ravens are black. Yep. The question is, how do you prove this?

The scientific method amounts to:

  1. Make observations
  2. Form an inductive hypothesis
  3. Experimentation to confirm or refute
The problem is, to prove the hypothesis, one has to observe as many ravens as possible, and confirm that they are all black. It is impossible to observe every raven - many of them have died, many of them haven't existed yet, and there could be ravens in inaccessible locations. However, each time we see a black raven, it tends to confirm the hypothesis that all ravens are black.

Yet, there is a part of this that is missing - only one side of the test has been performed. A -> B is equal to its contrapositive: not B -> not A. "If its a raven, then it is black" is equivalent to "if its not black, then it is not a raven". Therefore, every sighting of a non-black non-raven equally confirms the hypothesis. The red coffee cup on your desk is helping prove that all ravens are black.

Yes, it is silly, but by the laws of logic, if you accept an inductive hypotheses and confirmation by experiment, then every observation except the one that refutes the hypothesis confirms it - no matter how irrelevant.

There is a cute joke version of this, that uses an ambiguity in English to mask a logical error:

We want to prove all ravens are black. To directly confirm this, we would have to examine all ravens, which is not possible. Instead we show the contrapositive: all non-black things are not ravens. To prove it, we need only point to a single non-black object that indeed is not a raven, for example a white sheep. QED

(of course, this would work better if the above w/u had not made the reader more sensitive to the difference between "not all X are Y" and "all X are not Y".)

Actually here is a "proof" that all ravens are black, which uses a minor logical flaw in inductive reasoning to prove it.
We intend to prove that all ravens are black. To do this, we will reason inductively, and prove that for n ravens, all n ravens are black.

First, we prove P(1). To do this, we find a raven and confirm that he is black.

Now, we must show that P(n)->P(n+1), so we assume that for n ravens, all n ravens are black.

By our assumption, the first n ravens in the line of n+1 ravens are all black. Also, by the same assumption, the last n ravens in our line of n+1 ravens are black. Therefore, the 1st and the (n+1)th raven must be the same color as the middle (n-2), and therefore all n+1 ravens are black.

By the same measure, since P(n)->P(n+1), we have proven that all ravens are black.

Of course, the error in this proof can be found by examining the basis of induction, which is to show that P(1)->P(2)->P(3)->P(4)...

If you'll note, if we try to use the above argument for P(2), no overlap is created. Since we have no overlap, we have no basis to show that raven 1 is the same color as raven 2, since we can only assume that P(n) ravens are the same color, and P(n)=1 in this case.

The best way to do this is to define a raven as:

" A large black passerine bird (Corvus corax), similar to the crow, but larger. It is native of the northern part of Europe, Asia and America, and is noted for its sagacity.".
That is, if we take Webster 1913 as our ulimate authority. And who among us does not?.

Now, if it's not black, it's not a raven. By definition. QED

Until every raven is observed, the statement "All ravens are black" remains a theory. Theories are uncertain, otherwise we'd use the easier-to-spell word "fact". Confirmation or refutation by an experiment is also uncertain. Those Danish scientists that wrote that article in Science claiming they discovered a non-black raven; how do we know they weren't looking at a pigeon, or even more likely, were drunk?

This is where statistics comes in. Each experiment provides evidence to confirm or refute a theory (unless, of course, it's completely irrelevant). Observing a large number of black ravens provides significant evidence supporting the theory; observing a large number of non-black non-ravens provides a minuscule amount of evidence supporting the theory (unless you live in an alternative universe where almost everything is a raven). Observing a non-black raven, however, is strong evidence against a theory, especially if you're a credible witness.

What a respectable scientist (i.e., a scientist respected by a statistician) would do is:

  • attempt to estimate the conditional probability of the hypothesis given the evidence;
  • make a wild-ass guess of the prior probability of the hypothesis (i.e., will everyone laugh at me for trying to disprove it); then
  • apply Bayes' theorem to find the posterior probability (i.e., what everyone will think once they've read my paper).

A note to address dogboy's write-up:

The original point of this node was that in the context of inductive reasoning, logical consistency is equivalent to supporting evidence: a non-black, non-raven lends support to the claim that all ravens are black. My point is that not all evidence is equal and Bayes' theorem provides the best, and possibly the only consistent way, to update belief based on accumulating new evidence as it arrives, especially when dealing with support of theories. "Best", I'll admit, doesn't equate to the respect of statisticians.

I agree with dogboy's assertion that Bayesian reasoning isn't the most common tool of scientists and it's certainly not the only appropriate tool. A classical approach to the raven blackness problem might be to:

  1. Define the null hypothesis to be "there exists a non-black raven".
  2. Define blackness (probably in terms of luminosity and hue).
  3. Measure blackness over some sample of ravens. Convince everyone that your sample is unbiased.
  4. Calculate the P valueof your test.
  5. Claim victory if P ≥ 0.05.

I'll concede that this will get a scientist the respect of most statisticians (especially biostatisticians). But I'll also claim that it's not as convincing as a Bayesian argument and the classicist's distaste of a Bayesian's subjective probability isn't consistent with an almost religious acceptance of comparing P values to 0.05, an arbitrary number picked by Ronald A. Fisher in an attempt to eschew the equal arbitrariness of human belief.

The "all ravens are black" problem isn't really applicable to experiments, in the same way that common research problems in physics or biology might be. It's more akin to problems in astronomy, which makes high use of descriptive statistics. Descriptive statistics are useful for gleaning information from data but not for updating belief in theories; that's where Bayesian reasoning comes in.

Percepied makes some interesting comments about black ravens, theories, and statistics. All in all, a very good discussion. Here I merely add a bit of information to what is discussed in the last part of his/her write up where there is analysis of what a respectable scientist would do. Frankly, that is where there is a serious problems.

The problem is that Precepied suggests that a statistician would by definition be using Bayesian statistics to analyze data, or evaluate evidence. Nothing could be further from the truth. At the present time, my guess would be that statistical analyses that employ Bayesian statistics are, as a proportion of, say, journals that publish quantitative work, about 5% of all articles. Maybe less. Most statistical analysis relies on what we might call "classic" statistical analysis, which does not incorprorate Bayesian statistics.

A very quick review is in order, to point out to the reader what the major difference is between the two types of analyses. I'll keep it very short, as the main thrust belongs in another node. Classic statistical analysis tests the likelihood that a null hypothesis occurs by chance, and conversely, that the alternative hypothesis does not occur by chance. The null hypothesis usually assumes that some parameter is equal to zero. The value of some estimated parameter is compared to zero, and the difference between the two is tested for statistical significance.

Bayesian statistics does fundamemtally the same thing: Evaluating a null versus alternative hypothesis. The difference is in the assumption of what the null hypothesis tells us about the expected value of the parameter to be estimated. Bayesian statistics argues that that value should not be zero, but some other value, to be determined by the analyst. In theory, this makes a lot of sense. The problem is in the determination of those prior values. If not zero, what value then? This is where there are tremendous debates about the value of doing Bayesian analysis. In general though, there is great value in learning about, and possibly incorporating, Bayesian analysis into one's work. As long as the "priors" problem can be resolved somehow, of course.

So, in sum, a respectable scientist does not have to do any Bayesian statistics to be repsected by statisticians.

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A followup given Percepied's comments. Let me note that I didn't intend the write-up to be an attack on Bayesian statistics. In my opinion, the process is fascinating and is likely the appropriate approach for certain problems. The all ravens are black, in my opinion, is not one of those.

Why?

The all ravens are black problem, it seems to me , is the perfect kind of case to use for testing a null hypothesis H0:X=0, with the alternative being H0~=0. Null is all are black, alternative is at least one is not black. Using Bayesian stats, we would start with an assumption ( the prior value) that already suggests that not all ravens are black. That being the case, we wouldn't need to go any furhter... not all ravens are black by assumption, end of story.

In other cases, however, it might be best to use Bayesian stats. Suppose we have knowledge of a relationship between age and voting. As information is updated, we can change the priors, and move forward with the analysis. That would be fine. The all ravens are black problem is not one of this kind.

All said though, let me reiterate: This is not an attack on Bayesian stats, which I believe are fascinating.

I feel this is an appropriate place to tell the following quite well-known joke:

There was a train going north from England into Scotland, and in it were a Statistician, a Physicist and a Mathematician. Just after they passed the border into Scotland (with a sign up saying `You are now Entering Scotland') they looked out of the window and saw a black sheep in a field.

`Aha!' said the Statistician. `All Scottish Sheep are black.'

`Oh no', said the Physicist. `SOME Scottish sheep are black. Isn't that so?', turning to the Mathematician.

`I fear, gentlemen', said the Mathematician, `that you both go far beyond the data. All that can be said, and that with grave reservations, is that there exists in Scotland a field, in which there exists a sheep, AT LEAST ONE SIDE OF WHICH IS BLACK'.

Adding to the fun of black-ravens-as-knowledge-metaphor is Immanuel Kant's concept of the analytic-synthetic dichotomy.

Kant proposed that all such assertions can be categorized as either "analytic", in which the entity in question ("ravens") have an attribute by definition ("blackness"), or "synthetic", in which the attribute is not a defining characteristic of the entity is question, but rather added to the basic concept.

The issue becomes, in the case of making an assertion such as "All ravens are black", Kant would argue, that we are fundamentally just arguing about definitions. That is, either "ravens" are defined as being "black" birds (analytic), or "blackness" is an observed, non-defining characteristic of "ravens" (synthetic). Any such assertion comes down to how you (arbitrarily) define the entity in question. In other words, Kant would likely throw the question back and ask, "What do you mean by 'raven', and why is that definition valid?"

This dichotomy, though it potentially invalidates all human concepts, remains tenuously answered. The precise, "correct" definition of any given concept is approached through using Occam's Razor, but that method relies on a very debatable use of the criteria of "necessity".

(And I came to this node because I thought it was about Huginn and Muninn... "Thought" and "Memory"... hmm...)

The statement "all ravens are black" is false.

Just like the philosophical chestnut "all swans are white" was refuted when those pesky explorers discovered Australia (and along with it a refutation of almost every biological category) this claim has long been proven false through the judicious use of eyesight.

Yet again reality has thrown the epistemologist another curve ball with the albino raven.

An albino type raven crops up about one in every 10,000 eggs. Most of the time those birds will have only partial colour loss and so is technically a "leucistic" or "partial albino" raven. However in very very rare cases an entirely white raven will emerge. The last recorded incidence of a truly albino raven sadly died in 1997 in Port Clements Canada.1

The existence of white ravens might shock some of the more readily shockable philosophers, but to classicists, folklorists and corvidologists it will come as no surprise.

After all, mythologically speaking, ravens were originally white.

According to Greek legend, Apollo fell in love with the mortal Coronis and appointed a raven to look over her while she was pregnant with his child. When Coronis fell in love with Ischys and married him, the raven told the God about the wedding. After Apollo had destroyed the newlyweds in a fit of rage he turned all ravens black for being the bearers of bad news.

(This appears to be something of a habit for Apollo as in another story he turned all crows black for other message related reasons.)

Odin is famous for his two ravens Huginn and Muninn but he also seems to have a connection with white raven:

There are nights when the moon shines so brightly
and everything is ever so quiet.
Then Odin rides through the forest.
These are the raven nights.
When all the ravens are white and can speak,
and everyone can understand them 2

This could just be a way of saying that raven nights are very dark, but the more literal interpretation would be that during special nights when the raven are most capable they become white.

Some Native American tribes use the white raven to explain the position of heavenly bodies and the creation of fire:

The raven saw the sun, moon, stars and a burning stick being hoarded by a Goddess. Using its pure white nature the raven lulled the goddess into a false sense of security and stole the items. The raven placed the sun in the day sky and the moon and stars at night. However he could find no place for the fire stick and as he searched over the land his feathers became black with soot and his voice became hoarse. Eventually he had to drop the burning stick and the fire merged with the land, explaining why certain stones when rubbed together create fire.

There are many other stories about ravens getting their colour. Some say that another bird coloured the raven in black with ash, and another says that he flew at night for too long and soaked up the colour. In these and many other fables the raven sacrificed its whiteness to help either the Gods, humanity or other animals flourish.

However some Native Americans consider the re-appearance of a white raven to be a portent of Armageddon3 and so perhaps we should join the philosophers and be less than happy to see these albino birds:

"At the end of the world there shall be seen a white Raven as a sign that the world is coming to an end, that will be the last of it."




On a personal note, the reason why I'm interested in albino birds is that before I was born when my mother was young she decided to do a school project on one of my grandfather’s hobbies; ornithology

My grandfather however was quite famous for making up nonsense stories to entertain children. For example if we went walking in the woods he would tell me that he knew how to gather milky ways from the secret milky way trees. Then after performing a slight-of-hand he seemingly plucked a chocolate bar from the branches.

Therefore when my mum produced a report on a rare albino blackbird which had nested in her garden, a few people thought that he was up to his old tricks. Certainly the school children thought it was funny that my mum had apparently been fooled by this tale of a white-blackbird.

However a school trip and a local newspaper article later my relatives' claims were vindicated and soon my grandmother’s kitchen was full of local twitchers trying to catch a glimpse of the extraordinary bird.


1 Picture of albino raven in the event that you think I’m talking malarkey
2 A collection of raven related legends
3 white raven as portent of doom
4In Icelandic the saying "sjalds_enir hvitir hrafnar" meaning an infrequent occurrence similar to our "once in a blue moon" literally translates as "white ravens are a rare sight"
An obsessive collection of raven facts quotes etc
this amazing article found by bookreader shows that white ravens exist and are alive today when not one but THREE were found in a churchyard in County Durham!
If anyone remembers the rhyme for ravens it goes: "One for bad news Two for mirth, Three is a wedding, Four is a birth." If the bride is superstitious then that’s the church to get married in!

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