«

»

Jul 13

The Paradox of the Ravens and Hypothesis Testing

Generally speaking, when an atheist views the world, they tend to do so with a respect for hypothesis testing. We find something admirable in the notion that you can come up with predictions, test them in the real world, and the end results affect your view. Since very few things are certain, these outcomes generally either raise or lower probability.

Hypothesis testing  is a central component to many theories about why science has been successful, but there is a popular counter argument. It is based on something called The Paradox of the Ravens. I wrote a paper on this strange paradox once that I thought I’d share in case it may aid you in defending hypothesis testing against a clever opponent. This piece is meant as an introductory level explanation of the problem and why I think it’s not really a problem at all. Enjoy!

 

Observing Green Frogs: Why we should accept Hempel’s paradoxical conclusion

  

Introduction

Carl Hempel’s Paradox of the Ravens argues that observing a green frog actually confirms the hypothesis ‘All ravens are black.’ But why would observing a green frog confirm any knowledge about ravens? It strongly disagrees with our intuitions, but this is the case, according to the paradox. The paradox has generated considerable controversy among philosophers and scientists since its introduction by Hempel. I will defend the counterintuitive result to argue that, yes, observing a green frog can confirm knowledge about ravens.

 

Observation and Confirmation

Given certain skeptical problems about knowledge due to our epistemic limitations, it is very difficult to prove a theory or hypothesis correct. Instead, we will often say that experimental results or observations will confirm or disconfirm a particular hypothesis. If a hypothesis is confirmed, that means the probability of it being correct is raised by the observation, however slightly. Conversely, the probability is decreased when a hypothesis is disconfirmed.

We will deal primarily with confirmation here. Your probability is affected because an observation should limit your sample space in some way when confirming a hypothesis. Ideally, if an observation is confirmed, then you eliminate certain alternative explanations. Consider a murder case with three suspects. For this example, we will take it as a given that we know it must be one of these three suspects. The prior probability is as follows:

Suspect A: 1/3

Suspect B: 1/3

Suspect C: 1/3

Let’s say that the detective has a hypothesis that Suspect C is guilty. A consequence of this hypothesis is that Suspects A and B are not guilty. Our experiment will test for the effects of firing the gun, so the guilty person should have residue on their hands from the gunpowder. Even though the detective has a suspect in mind, he decides that he will not let his bias affect experiment and he tests all three in order.

The first test shows that Suspect A does not have residue on his hands. The probabilities are now as follows:

Suspect A: 0

Suspect B: 1/2

Suspect C: 1/2

The probability of the hypothesis that Suspect C will have gunpowder residue on his hands has increased from 1/3 to 1/2. Or, you might say the sample space of possible alternatives has shrunk from three to two. This example in which we did not actually test the suspect named by our hypothesis will be relevant to how we solve the paradox of the ravens.

 

Logical Equivalence

Before giving the paradox, we should understand logical equivalence. To say that ‘All F’s are G’ is equivalent to saying that ‘All non-G’s are non-F.’ Many find the abstract formulation of this argument confusing, but it should be clear with a more concrete example. We can say that ‘All numbers ending in 2 (F) are even (G).’ So, the equivalent then is ‘All numbers that are not even (non-G) do not end in 2 (non-F)’. This is clearly true.

Or, to return to our murder case, we might say our initial hypothesis once the experiment was decided was something like, ‘Suspect C [and only Suspect C] will have residue on his hands’. So, our logical equivalent would be ‘Any person without residue on their hands will not be suspect C’. We saw that this was true in the above case, at least when Suspect A was tested.

 

The Paradox of the Ravens

The case with examining one of the other suspects seems straightforward, but what happens when we introduce something that seems completely unrelated to the hypothesis? This was the goal of Hempel’s Paradox of the Ravens.

The Paradox

  1. If an observation O confirms a hypothesis H, then O confirms anything logically equivalent to H.
  2. Observation of an F that is G confirms ‘All F’s are G’
  3. ‘All non-G’s are non-F’ is logically equivalent to ‘All F’s are G’
  4. Observing a non-black non-raven confirms ‘All non-black things are non-ravens’ [From 2]
  5. ‘All non-black things are non-ravens’ is logically equivalent to ‘All ravens are black’ [From 3]
  6. Thus, observing a green frog confirms ‘All ravens are black’ [From 1,4,5]

The conclusion is certainly an odd one if our intuitions are to be trusted. One would expect that to confirm something about ravens, observing non-ravens won’t do the trick. Yet the construction of the argument is valid and, given what we have seen about confirmation and logical equivalence, the premises seem sound.

So what do we do with the paradoxical conclusion? We should accept it. As I hope to show, observing a non-black non-raven does confirm ‘All ravens are black’ ever so slightly.

 

Limiting our Sample Space

I think much of the confusion arises because of the very small increase in probability that occurs upon observing a green frog. While our intuitions can easily spot relatively large increases in probability, as with the murder case above, we do not so easily spot relatively small increases. Hence, we feel that the result is counterintuitive.

If we return to the example of the murder suspects, we can hopefully recognize that limiting our possibilities made it easy to see exactly why the probability was raised. Recall that it was raised without testing Suspect C. It would be nearly impossible to construct an accurate model of all black and non-black things in our world, let alone our universe, because the numbers are too large and fluctuate regularly. However, we can again simplify our sample space to clearly illustrate how the conclusion of the paradox works.

Let us consider a possible world W1, which is illustrated in Figure 1. W1 consists of 15 objects, and only these 15 objects. These can be divided into objects that are black and objects that are white (that is, non-black objects). There are 10 white objects and five black ones.

 

Figure 1

 

An explorer decides to visit W1 and discern things from the objects in it. She comes upon her first object and it is a black raven (noted with an asterisk in Figure 1). From this, she generalizes that perhaps all ravens are black. We do not know at this point if any of the other objects on W1 are also ravens. Perhaps more of the black objects are ravens and perhaps some or all of the white objects are also ravens. Every object could be a raven! Given this, even observing every black object in W1 will not provide a proof.

 

So, her hypothesis needs further confirmation. Our explorer continues to wander until she comes upon a white handkerchief. She suddenly has an idea; since she already knows there is at least one black raven, she can observe all the white objects in W1 and determine the truth of ‘All ravens are black.’ She sets out on her mission and eventually does examine all 10 white objects. None of them turn out to be ravens. This shows with absolute certainty that ‘All ravens are black’ in W1. Perhaps this can best be illustrated by Figure 2.

Figure 2

As you can see, she has crossed out every white object. The only objects left to observe in W1 are black. We still do not know how many more of them, if any, are ravens. But we now know that any remaining ravens must belong to the class of objects that are black. Observing the first white handkerchief was the first step toward determining this.

 

Solving the Paradox

In the case of the detective investigating a murder, we saw that there were three possible suspects, which gave us a prior probability of 1/3 and was then raised to 1/2. In the case of W1, we might say that any of the white objects had an equal probability of being a raven since we knew nothing about this world except what had been described. We can now think of W1 as being 11 objects, since we only care about the one black raven observed and the 10 unknown white objects. So, when our explorer wandered around observing the white objects of W1, she knew that 1/11 objects fit the categories of both black and raven. We can consider the numerator here to be confirmed objects that fit her hypothesis and the denominator to be items of uncertainty. When she observed the white handkerchief, that fraction became 1/10 because one item about which she was previously uncertain had now become irrelevant in a sense. Later, when she observed five of the white objects, and none of them were ravens, then the fraction became 1/6. Finally, when she observed all 10 white objects, and none of them were ravens, the fraction was 1/1. This is actually not a probability at all because it is complete certainty.

In this case, we were increasing our probability with each individual observation. But what if we had a world with 500 objects of which 400 were white and we had at least one confirmed black raven? Well, we would increase by an equivalently smaller amount with each observation of a white object on this world. Now think of the actual world and how many non-black non-ravens there must be. With an estimate of 300 sextillion stars in the universe, the fraction of each observation must be unimaginably small. But remember that these small increases in probability simply make it less intuitively obvious. If we restrict our sample space, we can clearly see there is an increase in probability occurring.

 

Conclusion

Hopefully it is now clear that by observing non-black non-ravens, like a green frog, we actually do confirm our hypothesis in some way. We made the example easier to see by limiting our sample space, but in the real world the degree of confirmation would miniscule. This would of course not be an effective way to do science, since our possibilities are not restricted like those of W1, but we can at least maintain that our confidence in the logic of hypothesis confirmation is not affected by this paradox.

We saw that observing non-black non-ravens did not confirm our hypothesis in the normal way by adding things that met the description of ‘all ravens are black’; rather, it confirmed our hypothesis by eliminating items of uncertainty, which potentially could have disconfirmed our hypothesis. Observing one green frog is simply checking off one item from a list that would seem infinite to us.

 

[Sources: Black, Andrew. The Authority of Science.]

Similar Posts:

6 comments

Skip to comment form

  1. Ryan

    In other words: In a universe that contains a finite number of objects, we can raise the probability of a hypothesis like “All ravens are black” being true through the process of elimination.

    I’m not sure all the text was necessary to establish the point unless I’m missing something. Also, for a while I was a bit confused by the “premises” of the paradox as you presented them because you used the term “confirms.” I eventually realized that you were using the term in the sense of “supports,” which is a perfectly acceptable definition, but it is very often used in the sense of “proves,” which would not work with the argument as presented, since premise 2 would be false and the conclusion really would not make sense.

    Perhaps these (small?) criticisms would be of no concern to the usual philosopher crowd, but it seems to me that much of the value of philosophy is lost on the public precisely because of faulty communication. We can’t enlighten the masses without catering considerably to the way they think and communicate. Many lack the patience to sit through posts like these and can easily be thrown off by formal language if they are used to the informal and familiar (the confusion over classical liberalism vs. American political liberalism comes to mind). Even I lack the patience to read, say, Heidegger. Of course, your post here is not nearly as impenetrable as his work, but I wonder: if we do not write for the average man who needs philosophy most, for whom are we writing?

    And with that, I step off my soapbox.

  2. Mike

    Yes, essentially that is correct. So even in our own universe, which I would say contains a finite amount of objects, observing a green frog raises the probability that all ravens are black. It’s just so small that it doesn’t really matter.

    Just to make sure it is clear, I am using confirmed to say that the prediction of the hypothesis was observed. The effect of this raises the probability the hypothesis is true.

    I agree it could be shorter. It was a paper for a former philosophy of science class so there was a word count goal (I barely made it).

    I’m sorry to hear you didn’t find it generally accessible. My goal was to actually make this paradox accessible to a non-philosopher and avoid resorting to Bayes to tackle it because I find those explanations too technical for the average person.

  3. Ryan

    I think the only real barrier to accessibility here is the length, which is excused because it was an assignment. (Word count goals are no friends to good philosophy.) The language is not complex even if some might be put off by the paradox itself. And what I said about formal language was merely something else that came to mind but is not necessarily applicable to your post. It’s entirely possible that I’m cynically underestimating the average non-philosopher, but at least, as you say, you did not use Bayes.

  4. tim d

    Saying things like…

    “…when an atheist views the world,…”

    is just a broad overgeneralization.

    Different folks have different thinking patterns…some very smart folks are not “logical thinkers” but depend more on intuitive logic. Whether or not they are “believers” is typically more about history than sense.

    Some folks are atheists and arrogant at the same time, others are humbled by their knowledge of the truth.

  5. tim d

    i forgot to point out

    2.Observation of an F that is G confirms ‘All F’s are G’

    is clearly bogus logic

  6. Mike

    Tim,

    I agree it can be an overgeneralization and I do try to keep my posts based on specific claims or arguments. My goal here is to counter the paradox, so hopefully the extra narrative doesn’t muddy the waters too much.

    Regarding (2), I think I should clear up that I mean “confirm” in a particular sense. I mean it as in it increases the probability the initial statement I true. I don’t mean it in the sense of “proves.”

Leave a Reply

Your email address will not be published. Required fields are marked *

You may use these HTML tags and attributes: <a href="" title=""> <abbr title=""> <acronym title=""> <b> <blockquote cite=""> <cite> <code> <del datetime=""> <em> <i> <q cite=""> <s> <strike> <strong>

Notify me of followup comments via e-mail. You can also subscribe without commenting.