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The raven paradox and the confusingly large size of negligibility

/ 7 min read

non-technical

1. Logical premises, illogical conclusions

The paradox builds from these apparently innocent premises:

  1. The propositions “all ravens are black” and “all non-black things are non-ravens” are logically equivalent, meaning one is true if and only if the other one is true.
  2. For each black raven one finds (without finding a raven of a different color) one can be slightly more certain (if only marginally) that all ravens are black.
  3. Equivalently, for each non-black non-raven thing one finds (without finding a non-black raven) one can be slightly more certain that all non-black things are non-ravens.

And now perhaps you can imagine what’s coming. If you look at the wall, which is not black and not a raven, you can be a little bit more sure that all non-black things are non-ravens, and equivalently, that all ravens are black. In fact, it seems like everywhere you look you will see evidence that all ravens are black, without the need to see a single raven.

Clearly there’s something wrong here, or is there? Where would the mistake be? As one examines the issue more deeply, it becomes clearer that the paradox is not such. In fact, it turns out to be a rather intuitive result, only obscured by the lack of quantitative precision of the statements involved.

2. Getting around the paradox with the aid of quantity

How so? The above argument could be used not only to grow more certain that all ravens are black, we could have said all ravens are green, or all unicorns are blue.

Let’s get unicorns out of the way. In logic, a universal statement is said to be true (vacuously true) whenever the antecedent cannot be satisfied, so any proposition of the kind “all unicorns are…” is vacuously true. Therefore, there’s no problem if we use the above procedure to grow more certain that all unicorns are blue, yellow or pink, because in a way they are (in other words, you could say that the only things that can hold mutually exclusive properties are the non-existing things).

Now back to ravens. A generally useful thinking tool consists in considering extreme cases, as it often helps make certain features more apparent; so let’s give it try. If we saw 99.9% of evenly distributed1 ravens and all happened to be black, we could be quite sure that all ravens are black. Similarly, if we saw 99.9% of evenly distributed non-black things, and there were no ravens there, then we could be quite sure that ravens, if they exist, must be on the bucket of black things.

So now both propositions lead to the same sensible result, where did the paradox go? The key is that now we are considering in each case a fractional amount of the total population, so that the evidence in each case is more comparable. If looking at 160,000 black ravens would cover 1% of the total population (there seem to be approximately 16 million ravens on Earth), looking at 160,000 non-black things is probably not even 0.00000001% (there’s room for adding quite a lot more 0s but it stops making sense at some point) of the total population, so in every respect that evidence could be considered negligible and outright ignored.

In summary, the solution to the paradox is quantitative, we should be more certain that all ravens are black after seeing a white wall, but by such a tiny amount that it really makes no difference, hence our minds intuitively do not bother about this.

Physics can provide another example of how taking into account negligible amounts breaks our intuition. Fluids are made of randomly moving atoms. Then there’s a non-zero probability that in a glass of water at room temperature the slow moving atoms concentrate on an area that spontaneously freezes, or that all the oxygen atoms in your surrounding move away from you as you are about to catch your next breath. Why does this sound ludicrous? Because the probability is so low that it has almost certainly2 never happenned in the entire history of the universe.

Perhaps a good way to summarize the discussion would be to say that between non-zero and non-negligible amounts there’s a whole lot of space, and it’s in playing with the size of this space that the apparent paradox comes to existence.

3. Reflecting on lessons and the way ahead

A nice thing about this paradox is that along the way it showcases some problem solving tools that are nice to have in mind.

The first is reframing, often different statements can be equivalent in their meaning but not in the effort it takes to demonstrate them. Hence, being able to come up with as many reformulations as possible is sometimes the most important step in problem solving. In logic and math it’s common to use the contrapositive (the equivalent statement with negated conditions as in “non-black things are non-ravens”) to find gentler paths towards solutions, and this can be generalized to other fields. In this case contraposition is actually used the opposite way, that is, to complicate things instead of simplify them; but perhaps the degree of complication that is able to create from an apparently innocent statement can convey the notion of the degree of simplification it can bring about when used in the right direction.

The second is the use of extreme cases to simplify the problem. One instance of this can be considered the initial reductio ad absurdum where the paradox is pushed to its ultimate consequences to show it’s non-sensical and requires further investigation. Extreme cases help identify the points of failure in an argument, but also its points of success, as the other instance, namely considering a sample size of 99.9%, shows.

Thirdly, and probably most importantly, our discussion highlights the importance of quantifying certain propositions and being precise in this regard. The origin of this paradox is in using vague terms as “a little bit more certain” that can mean many things, but hardly is interpreted as an absolutely negligible amount (and rightly so, because what would be the point of talking about negligible amounts?). Hence, proper reasoning requires quantification3.

Now, to conduct a more formal exploration of the raven paradox we will resort to the queen of quantity: maths. It turns out that this is also a great excuse to introduce Bayesian inference, along with its many applications and some discussion on the difficulties that may appear in real life implementations. If you find this interesting, you may want to take a look at this post.

Footnotes

  1. This evenly distributed condition is key, and hard to fulfil in reality, as we would be bounded by geographical constraints which makes our observations related among themselves and therefore not as informative as a truly random sample. This has little relevance for the existence/non-existence of the paradox we are discussing, but it’s always nice to have these kinds of caveats in mind. ↩

  2. Technically we cannot say certainly because the probability is non-zero, but it’s again so low that for all practical purposes can be considered a sure thing. ↩

  3. At least if we don’t restrict ourselves to the black and white realm of logic and we venture into the much more common realm of beliefs and probabilities that are made of shades of gray. ↩