The scientific method is only a few hundred years old. This continues to amaze me. It
seems so obvious, now, that you should go and test your theories and, if necessary,
revise them. But for much of human history, coming up with a “theory” was merely about
story-telling and sense-making, not about making quantitatively accurate predictions.
Then again, the scientific method is not set in stone. Scientists and philosophers both
are still trying to understand just how to identify the best hypothesis or when to discard
one. This is not as trivial as it sounds, and this difficulty is well illustrated by
the Raven Paradox, which I want to talk about today.
The Raven Paradox was first discussed in the 1940s by the German philosopher Carl Gustav
Hempel and it is therefore also known as Hempel’s paradox. Hempel was thinking about what type
of evidence counts in favor of a hypothesis. As an example, he used the hypothesis “All
ravens are black”. If you see a raven, and the raven is indeed black, then you’d say
this counts as evidence in favor of the hypothesis. So far, so good.
Now, the hypothesis that all ravens are black can be expressed as a logical statement in
the form “If something is a raven, then it is black.” This statement is then logically
equivalent to saying “If something is not black, then it is not a raven.” But once
you have reformulated the hypothesis this way, then anything not black that is not a
raven counts in favor of your hypothesis. Say, you see a red bus, then that speaks for
the hypothesis that ravens are black, because the bus is not black and it not a raven either.
If you see a green apple, that’s even more evidence that ravens are black. Yellow post-its?
Brown snails? White daisies? They’re all evidence that ravens are black!
To most of you this will sounds somewhat nuts, and that’s what’s paradoxical about it.
The argument is logically entirely correct. And yet, it seems intuitively wrong. This
is not how we actually go about collecting evidence for hypotheses. So what is going
on? Do we maybe not understand how science works after all?
Hempel himself seems to have thought that our intuition is just wrong. But the more
commonly accepted explanation is today that our intuition is right, at least in this case.
This explanation has it that we think black ravens are better evidence for the hypothesis
that ravens are black than non-black non-ravens because there are more non-black non-ravens
than there are black ravens, and indeed we have seen a lot of non-black non-ravens in
our lives already. So, if we see a green apple, that’s evidence, alright, but it’s not
very interesting evidence. It’s not very surprising. It does not tell you much new.
This argument can be made more formal using Bayesian inference. Bayesian inference is
a method to update your evaluation of the probability of a hypothesis if you get more
information. And indeed, for the raven paradox the calculation seems to be showing that the
non-black non-ravens *are evidence in favor of the hypothesis, but black ravens are better
evidence. They help you gain more confidence in your hypothesis.
But. The argument from Bayesian inference expects you to know how many non-black non-ravens
there are compared to ravens. You might estimate this to be a large number, but where do you
get the evidence for that number from? And how have you evaluated it? What do you even
mean by a non-black non-raven. Come to think of it, just how do you define “raven”?
And what does it mean for something to be “black”? And so on. You can debate this
endlessly, if you want.
But you know me, I don’t want to debate this endlessly, I just want to inspire you
to think about this paradox for a moment and maybe confuse some other people with it. Thanks
for watching, see you next week.