Raven paradox

An apparent paradox in epistemology, which raises the question of under what conditions observations can actually confirm a general hypothesis.

In the context of the topics covered on this website, this paradox is also of interest because it highlights a problem concerning the equivalence of logical propositional forms, which would otherwise be difficult to grasp.

• Hempel’s paradox

The paradox, originally formulated by Carl Gustav Hempel in the 1940s, is based on the following statement, which at first glance appears quite innocuous:

All ravens are black. (A ⟶ B)

We are dealing here with a positive universal proposition, or, depending on one’s perspective, a positive (material) conditional proposition. For now, however, we are less concerned with the logical form and more with the question of how we can determine whether this statement is true – in other words, on the basis of what information we can confirm that all ravens are indeed black.

Of course, it would be ideal if we could simply collect all the ravens to see whether they really are all black. However, this is at the very least very difficult when it comes to wild birds, and for many other things to which such statements might apply, it may even be completely impossible. So we need a different approach.

Instead, let’s simply look at the ravens we happen to come across: the more black birds we encounter, the more likely it becomes that the universal claim is actually true. Conversely, however, a single non-black individual is enough to contradict the claim. If such an animal were to exist, the claim would be proven false and would therefore be untrue.

In other words: every black raven we observe supports the hypothesis that the statement “All ravens are black” is indeed true.

For the second step, we now convert the statement above into an equivalent negative statement using a Contraposition or Modus Tollens:

If something is not black, then it is not a raven. (⌐B ⟶ ⌐A)

This transformation is valid, and the two statements are in fact logically equivalent; that is, both statements have the same truth value under all circumstances.

Hempel now argues that it follows that any observation of non-black non-ravens – such as a yellow post van or a blue vase – must necessarily also support the original claim that all ravens are black.

Most people will no doubt reject this at first and intuitively regard the observation of “non-ravens” as irrelevant to the question of whether all ravens are indeed black. From a logical point of view, however, it is correct at the outset: if both statements are logically equivalent, the conclusions that can be drawn from them must also be the same.

A well-known counterargument to this paradox points out that although the two statements are logically equivalent, they refer to different-sized reference classes (extensions), which makes the sample size relevant.

In other words, one can concede that the sighting of non-black non-ravens does indeed support the statement “All ravens are black”, but given the vastly greater number of non-ravens in the universe compared to the relatively small number of ravens, this support is infinitesimally small, meaning that the relevance of such a sighting is, in fact, practically negligible.

This response does not fully resolve the problem – it is easy to imagine statements in which a negation does indeed refer to a smaller set, such as: “All atoms are small” – but it does highlight the fact that, even though the two forms of the statement are formally logically equivalent, they may nevertheless have different properties that can become relevant in certain situations.

Even though different forms of logical statements can be derived from one another – not only, but particularly if they are completely logically equivalent, as in Hempel’s paradox – they still have differences. If these are ignored, unexpected results may arise.

Some of the distinctive properties of propositional statements, particularly categorical ones, are well documented in logic. These include, for example, whether terms are distributed or whether a propositional statement implies existence. Other aspects, however, only become apparent in specific applications.

The specific logical problem here lies in the difference between the conclusions that can be derived from negative and positive universal propositions. Although this difference is already suggested in several well-known logical fallacies (see, for example, the fallacy of exclusive premises), it is by no means as formalised as, for instance, the concepts of distribution and existential implication.

Another important point that is easily overlooked: in fact, not all ravens are black! As well as rare albino varieties, there are also individuals that, for various reasons, have plumage that is rather grey or reddish-brown. These are rare exceptions, but if we are to apply a strict interpretation of logic, then a single counterexample is enough to disprove the universal statement¹⁾.

At the same time, however, we can see from well-documented logical fallacies (see, for example, the fallacy of eliminative induction) that methods of logic which work well within formal systems can fail in the face of the complexity of reality if they are applied to the latter uncritically. It is quite conceivable that Hempel has “merely” found a good example of a similar problem here.

It would thus be worth considering, to say the least, whether an overly strict application of logic is really the right approach in such cases. At the very least in real-life situations – that is, outside of thought experiments – one should adopt a pragmatic approach and take the complexity of the situation into account right from the initial statement. A formulation such as: “Ravens are typically black” would, in this specific case, not only describe reality more accurately, but also eliminate the temptation to apply methods of propositional logic outside their scope of application, where they can lead to unexpected (and possibly unrealistic) results.

• Fallacy of eliminative induction – incorrect application of an inductive method
• Fallacy of exclusive premises – a formal fallacy in syllogistic logic
• Vacuous truth – a logically true statement that has no informative value

• Raven paradox on Wikipedia
• Carl Hempel § The Paradox of Confirmation on Stanford Encyclopedia of Philosophy