Paradox of the Ravens

The paradox of the ravens was formulated by philosopher Carl Hempel. It is essentially a paradox of confirmation.

For any stated generalization, an instance of its truth confirms the generalization. For instance, if I made the generalization that all former and current U.S. Presidents are male, then any instance of a male president would confirm that generalization. Let us represent this principle, that a generalization is confirmed by any instance of its truth, as G1.

The other principle necessary to illustrate the paradox of the ravens is that if two hypotheses can be known to be logically equivalent, then any data that confirms one of them confirms the other. This can be known to be true, as if the statements have identical truth tables, any data that confirms one statement will confirm the other. This principle will be represented as E1.

Now let's imagine two equivalent hypotheses. First, "All ravens are black" (R1), and second, "All non-black things are non-ravens" (R2). These statements can be known to be logically equivalent by the Law of Contrapositives (p>q)≡(-q>-p). Since these statements are equivalent, any data that confirms one of them should confirm the other. This is where our paradox comes in. Statement R2 can be confirmed by the existence of anything that is non-black and also not a raven (e.g. almost anything). Therefore, by E1, we should expect statement R1 to also be confirmed by the same data. This is, on the face of it, an absurd notion. Why should the statement "All ravens are black" be confirmed by the existence of, say, a white piece of paper or a red apple?

There is no immediately obvious answer to the paradox. All of the premises seem to be true, and the reasoning valid. Therefore, the only way left to deal with the paradox must be to accept the conclusion. This is not necessarily as ridiculous as it seems. Simply because data that confirms one hypothesis confirms the other, it doesn't mean that the data confirms both of them to the same degree. In a way, the existence of a non-black thing that is not a raven does confirm the hypothesis that all ravens are black.

Think about it this way. If we actually managed to seek out and find every single non-black thing in the world, and were able to determine that none of them were ravens, then we would prove that all ravens are black. (This is probably not the most efficient way to prove this hypothesis.) Therefore, the existence of each individual non-black non-raven confirms R1, though not nearly to the same degree as the existence of a black raven does. This is simply because there are far more non-black things in the world than there are ravens.

Introduction

The Raven Paradox was proposed by Carl Hempel in the 1940s to illustrate a problem where inductive logic appears to violate intuition. Put simply, it is a paradox of confirmation and concerns applying the principle of induction to a theory. That is, do observations that are consistent with a theory increase the probability of that theory's truth? Hempel's simplistic example theory was that all ravens are black. In order to attempt to show that this theory is correct, one could go and observe a large number of ravens and then determine what colour they are. Each black raven that is detected lends weight to the theory that claims all ravens are black. However, the logical equivalent of the theory "all ravens are black" is "all non-black objects are not ravens". According to this particular formulation, observing a non-black non-raven object would back up the original hypothesis. However, at first it appears absurd that observing a green cabbage should make it more likely that all ravens are black.

Non-black objects

As it is logically incontrovertible that the two hypotheses are equivalent, either the observation of green cabbages (or apples or yellow bananas and so on) does confirm the theory that all ravens are black or it does not. Although it would intuitively appear that these data are completely irrelevant to the hypothesis, Hempel's thought experiment seems to show that this is not the case. However, there are other possible observational outcomes that do not necessarily confirm the theory. For example, a single white raven would falsify the hypothesis irrespective of the number of black ravens and green cabbages that had previously been observed or are yet to be discovered. It can be argued that the probability that the theory is true can be increased by repeated failures to falsify it. Using a certain method under which it is likely that non-black ravens will be discovered provides stronger evidence for the theory than one which has a small chance of detecting counterexamples.

An even better way to ascertain the veracity of the theory would be to check every single object in the universe to see whether there are any ravens that are not black. Leaving aside practical concerns, this method would establish one way or the other whether the hypothesis is true or false. An imperfect test would be to randomly sample objects from a set of all the objects in existence. Every object sampled that turns out to be a non-black raven counts as weak evidence in favour of the theory. Similarly, observing green cabbages in the same sample also acts as evidence for the hypothesis. This is because, given a constant number of green cabbages in existence, they are slightly more likely to turn up in the sample given that there are non-black ravens and all other variables remain the same. Although this may appear to be counterintuitive, it should be remembered that the strength of confirmation provided in this case is reliant upon the proportion of non-black ravens expected in the sample given that non-black ravens are there to be observed. The confirmation given by an observation depends on the size of the sample.

To improve this proportion and hence the degree of confirmation, the sampling ought not to be completely random. For example, if only the class of non-black objects were to be sampled no possible non-black ravens would be excluded from the sample, yet the class size would be smaller and hence the confirmation greater. Failing to see a raven among them would show that the hypothesis is likely to be correct. To increase the strength of the results a smaller class could be considered such as all the ravens in the universe. Each non-non-black raven in the sample would provide evidence in favour of the theory and hence the intuitive idea that black ravens grant powerful confirmation is arrived at.

Confirmation?

It is important to realise that sampling non-black objects and showing that they are not ravens provides confirmation whereas sampling non-ravens and showing that they are not black does not. The predicates are not directly exchangeable because sampling the population of non-ravens things does not provide the opportunity to observe non-black ravens and hence disprove the hypothesis. Sampling non-ravens or even all black objects does not confirm the hypothesis to any degree. Should counterexamples exist, the only way of detecting them is to choose a suitable sample. Testing the set of non-ravens gives no data about the set of ravens with which we are concerned.

Hempel uses Nicod's Criterion to decide whether or not an observation has confirmatory power and defines is as follows: "Whenever an object has two properties C₁ and C₂, it constitutes confirming evidence for the hypothesis that every object which has the attribute C₁ also has the attribute C₂". This is clearly not a good criterion as observing cabbages provides no information about the existence of non-black ravens. It is only through observing objects that could be non-black ravens that falsification is possible and it is argued that this possibility is necessary for confirmation. Hence, it may be believed that observing a green object and showing that it is a cabbage is confirmation for the hypothesis whereas looking at all the cabbages and noticing that they are all green does not.

Conclusion

From these results, it can be argued that the apparent existence of a paradox is misleading. Observing a green cabbage can quite reasonably be taken to provide evidence in favour of the theory. However, given the difference in sample sizes that we would expect, much stronger confirmation is to be had by finding a raven to be black than by finding a non-black object to be a non-raven. The degree of confirmation from the latter is so weak as to be unworthy of notice for all practical purposes. The sole problem with this is that there may be examples where it infeasible to estimate relative class size. In addition, it has been shown that Nicod's Criterion is incorrect because there is no data for confirmation to be gain by sampling objects that are by definition cannot be non-black ravens. Hence, black ravens themselves are not always evidence in favour of the proposed theory. While it appears strange that observing green cabbages increases the likelihood of the hypothesis being true, this can actually be the case under certain circumstances.

Sources:

• Hempel, Carl. Philosophy Of Natural Sciences (1966).
• Hempel, Carl. "Studies In The Logic Of Confirmation", Aspects Of Scientific Explanation (1965).
• Mackie, J. L. "The Paradox Of Confirmation" The British Journal for the Philosophy of Science, 13, pp. 265-277 (1963).