Does seeing a white tennis shoe make it more likely that all flamingos are pink?

Introducing the Accessibility Principle

UNDERSTANDING HEMPEL’S PARADOX

In the mid-20th century, philosopher Carl Gustav Hempel introduced a paradox that came to be known as ‘Hempel’s Paradox’ or the ‘Raven Paradox’. The paradox begins with a seemingly simple and clear premise: If the hypothesis is that ‘all ravens are black’, then any observation of a black raven should help to support the hypothesis.

However, Hempel pointed out that this statement is logically equivalent to the statement: ‘Everything that is not black is not a raven’. Hence, any observation of a non-black, non-raven object, such as a white tennis shoe, should also help to support the hypothesis.

Yet it feels strange to believe that seeing a white tennis shoe should serve to increase our belief that all ravens are black.

HEMPEL’S PARADOX AND THE COLOUR OF FLAMINGOS

Now, let’s apply this principle to another statement: ‘All flamingos are pink’. This proposition is logically equivalent to: ‘Everything that is not pink is not a flamingo’. By Hempel’s argument, observing an object that is not pink and not a flamingo, such as a white tennis shoe, would provide evidence in support of the hypothesis that all flamingos are pink.

From a formal logic perspective, this argument makes sense. However, our intuition may still find this hard to accept, mirroring the original conflict inherent in Hempel’s Paradox.

TESTING THE HYPOTHESIS

In conventional hypothesis testing, we would go out and find some flamingos, verifying if they are indeed pink. But the Raven Paradox suggests that we could conduct meaningful research by simply looking at random non-pink things and checking if they are flamingos. As we collect data, we increasingly lend support to the hypothesis that all non-pink things are non-flamingos, equivalently that all flamingos are pink.

While this approach holds up logically, it does have its limitations. Considering the vast number of non-pink things in the world compared to the population of flamingos, the hypothesis can be much more confidently validated by sampling flamingos directly. Hence, although Hempel’s Paradox does not contain a logical flaw, it is not an efficient or practical method for testing the hypothesis.

THE ACCESSIBILITY PRINCIPLE (OR OBSERVATIONAL LIKELIHOOD PRINCIPLE)

Suppose we have two hypothetical species—one is a type of bird that frequents populated areas (Species A), and the other is a rare kind of lizard that lives in remote, inaccessible areas (Species B). If both these species are unobserved, it’s more likely that Species B exists rather than Species A, because Species B is less likely to be observed due to its habitat even if it exists. In contrast, Species A should have been observed if it were real due to its frequent presence in populated areas. I term this the ‘Accessibility Principle’, or alternatively the ‘Observational Likelihood Principle’. These terms suggest that the likelihood of an entity’s existence depends on its observability. This aligns with real-world scientific practices, where the absence of evidence is not always evidence of absence, particularly when dealing with hard-to-observe phenomena.

So, let’s take the propositions in the thought experiment in turn. Proposition 1: All flamingos are pink. Proposition 2 (logically equivalent to Proposition 1): Everything that is not pink is not a flamingo. Proposition 3 (the ‘Accessibility Principle’): If something might or might not exist but is difficult to observe, it is more likely to exist than something which can be easily observed but is not observed.

Following from these propositions, when I see two white tennis shoes, I am ever so slightly more confident that all flamingos are pink than before. This is especially so if any non-pink flamingos that might be out there would be easy to spot. And I’d still be wrong, but for all the right reasons.

CONCLUSION: THE OBSERVATION PARADOX

In summary, Hempel’s Paradox is an intriguing clash between intuitive reasoning and formal logic. It forces us to confront the subtleties of hypothesis testing and belief formation. In this example, the paradox implies that we may gain a tiny bit more confidence in the hypothesis that all flamingos are pink if we observe a white tennis shoe. However, such indirect evidence should be considered in its appropriate context, not as a substitute for direct evidence. The key point of the paradox is instead to challenge our understanding of the meaning of evidence and to provide valuable insights into the nature of logical reasoning. Essentially, any hypothesis is always susceptible to new evidence that can strengthen support for it. In the case of the pink flamingo hypothesis, this applies whether it comes from observing a flock of pink flamingos or (to a much lesser degree) a pair of white tennis shoes. Until you see an orange flamingo, then you know otherwise!