Thursday 30 April 2020

Weight of Evidence

A very short educational piece, this time, on the subject of evidence.

Evidence is very important to genealogists. It is information that supports, or contradicts (as in 'evidence to the contrary'), some specific claim. When we make claims, such as those about parentage, then we need supporting evidence, and we also need to explain any evidence that appears to go against our claims.
But is that everything we need to know? Well, no; not all evidence carries the same weight. In order to illustrate this particular point, I want to introduce you to the 'raven paradox', sometimes known as 'Hempel's paradox' since it was formulated by philosopher Carl Gustav Hempel in the 1940s to illustrate a contradiction between inductive logic and intuition.

Hempel starts this paradox with the proposition 'all ravens are black'. This can be turned about-face to yield the equivalent proposition 'if something is not black then it is not a raven'.  The first of these is quite straightforward, and the sight of a black raven would be evidence supporting that proposition. However, that about-face proposition is less straightforward because the sight of anything other than a raven, and that isn't black, would be evidence for it. Hence, if you were eating a green apple then it's not black and it's not a raven, and so it supports the second proposition. The paradox is that something totally unrelated to ravens, or even to birds of any kind, appears to be evidence for that first proposition: 'all ravens are black'.

So what gives? Surely, the fact that you're eating a green apple, or wearing a red hat, or any number of unrelated observations, cannot really be evidence about the colour of ravens. Philosophers have debated this paradox ever since because that's what they like to do, but the answer is relatively simple. Yes, those observations really are evidence but their weight is so weak that they're effectively insignificant.

In order to understand what's going off, we need to consider the scope of the propositions. In this particular case, where the proposition is about discrete entities (ravens) and properties that are fixed (colour), then you can imagine sets of possibilities, but a more general scheme would involve abstract mathematical spaces of possibilities. Anyway, the spaces of possibilities for black-ravens, non-black-ravens, black-other, and non-black-other are vastly different in extent. Having an observation that supports non-black-other (an astronomically huge space) is insignificant compared to one that directly supports black-ravens, even though the propositions all-ravens-are-black and if-non-black-then-not-a-raven are logically equivalent. In contrast, if we observed just one instance of non-black-ravens (the space for which we've asserted to have zero extent) then it would be hugely significant.

The lesson, here, is that the same claim can be expressed in different, but logically equivalent ways, and this has a huge bearing on the significance of an item of information supporting the claim. The weight, or significance, of some evidence depends on the scope of the claim, and some cases — such as demonstrating beyond reasonable doubt that 'if something is not black then it is not a raven' — would be impractical to pursue. Putting things another way, the concept of 'sufficient evidence' depends on the scope, or the number of possibilities, covered by the claim.