Raven Paradox Revisited
The Raven Paradox Settled to My Satisfaction is a pretty good post, but there’s a few things that we can note to make this problem even clearer:
We can simplify this problem so that there are only two colors (black and white) and two kinds of objects (ravens and laptops). The post sort of did this, but not very consistently. I just thought this was worth mentioning as non-black and non-raven are slightly more abstract and so slightly harder to reason with.
Actually, we can make this even more legible. Make the objects medicine and food; and the properties fresh and expired. It’s then immediately clear that if we want to check that [All the medicine is fresh], we can either check each item of medicine to see if it is fresh or we can check all the expired objects and see that none of them is medicine. We’re used to this kind of practical reasoning, so it’s much easier for us than dealing with objects and colours.
If we have simplified the problem so there is only one shade of black, then A: [All ravens are black] is equivalent to B: [The first raven is black] plus C: [All ravens are the same color] apart from the degenerate case. This makes it rather clear why A is more likely to be true when there are less ravens. In particular, if there’s only one raven, then we only have to worry about B since C is trivially true.
We tend to confuse the following [An observation of a white laptop is independent of the color distribution of ravens] with [An observation of a white laptop is independent of how many ravens of a particular colour we will observe]. The first is true apart from the restrictions imposed by the number of black ravens having to be integral, but the second is only true if we knew it was a laptop before this observation. If it could have been a raven then it can influence the number of ravens of a particular color by influencing the number of ravens in total.
We tend to expect some correlation between the color of animals. For example, we are quite ready to guess that all ravens are black after only seeing quite a small random sample. On the other hand, suppose that all the non-black things keep being non-living creatures, whilst many of the black things are non-raven living creatures. We might guess that being black provides an evolutionary advantage in this world and so guess that all ravens will be black without ever having seen a single one. The point is that, given particular priors, you may have additional evidence beyond merely the numerical reduction.
Hempel’s resolution from Wikipedia is worth highlighting as it subtly reframes the problem to make the assumptions more obvious. Consider the statement [All sodium salts burn yellow], with the contrapositive [Whatever does not burn yellow is not a sodium salt]. Burning some ice and finding it does not tell yellow would be evidence towards the contrapositive and hence also the original statement. This seems paradoxical, but consider if the chemical makeup was unknown at the start. If something doesn’t burn yellow and then we analyse it and discover it is a sodium salt then we would have disproved the hypothesis. By conservation of evidence, if we discover it is ice we would gain evidence for the hypothesis. It’s easy enough to calculate the amount of evidence using Bayesian techniques. From here, it’s easy enough to see that this observation only provides no information if we implicitly assume that we know the chemical composition (or the type of object in the raven problem).