The philosophy of science is mostly composed of unsettling revelations. Not only is it hard to say why inductive proofs should be good enough, it gets surprisingly murky even trying to define what counts as evidence for a hypothesis in the first place. The classic example of this is Hempel’s paradox, also known as the raven paradox. Let’s say Hempel, a German philosopher who was closely associated with the logical positivists, states the hypothesis that all ravens are black. What would count as evidence to support this? Most people would say that seeing any black raven would support the theory, while seeing any non-black raven would disprove it. Anything that is logically consistent with the theory provides support that it is true, and the same with the inverse. So far, so obvious, and Hempel didn’t disagree with any of this. He did point out something subtle, though.
Logically speaking, two statements are equivalent if they always give the same answer. “If something is human, it is mortal” is the same as “If something is not mortal, it is not human”. This is called contraposition. For any possible sample object X, you can apply both of those statements and always get the same evaluation. I am a human, so I am mortal. A poem is not mortal, so it is not human. In the case of Hempel’s raven hypothesis, this means that “all ravens are black” is logically equivalent to “all non-black things are not ravens”. That is the contrapositive of the original statement. It may be a bit of an inside-out way of looking at the situation, but it’s equally true. If we find a bright orange thing, we know it’s not a raven.
Above, however, we said that anything that is logically consistent with the hypothesis is evidence that the hypothesis is true. Finding an orange thing that is a traffic cone (and not a raven) is logically consistent with the hypothesis. So is the traffic cone evidence that all ravens are black? Logically speaking, yes, it is. As is every non-black, non-raven object in the universe. It turns out we’re surrounded at all times by overwhelming mountains of evidence that all ravens are black. The apple you had for lunch was evidence that all ravens are black. The Antikythera Mechanism in Athens is evidence that all ravens are black. The thousands of galaxies in the Virgo cluster are composed of quadrillions of solar masses of evidence that all ravens are black -- and they’re all so far away that a signal sent to them announcing the evolution of the corvid family wouldn’t even have reached the halfway mark yet. This is fairly counter-intuitive, to say the least.
There are several possible solutions to this conundrum. Hempel himself argued that the issue was in the assumptions we bring with us to the hypothesis. If you actually knew absolutely nothing about ravens and non-ravens, then you would more naturally consider every non-black, non-raven object as evidence of the hypothesis. Many other answers are focused on trying to parse out how much proof each piece of evidence provides. There are so many more non-raven objects in the universe than raven objects that every black raven provides astronomically more evidence for the hypothesis. This is a strong response, though in more complicated examples it gets hard to puzzle out exactly how much evidence something “should” be providing. But all the same, even the simplest assumptions about how scientific evidence works can quickly lead to weird conclusions.
Logically speaking, two statements are equivalent if they always give the same answer. “If something is human, it is mortal” is the same as “If something is not mortal, it is not human”. This is called contraposition. For any possible sample object X, you can apply both of those statements and always get the same evaluation. I am a human, so I am mortal. A poem is not mortal, so it is not human. In the case of Hempel’s raven hypothesis, this means that “all ravens are black” is logically equivalent to “all non-black things are not ravens”. That is the contrapositive of the original statement. It may be a bit of an inside-out way of looking at the situation, but it’s equally true. If we find a bright orange thing, we know it’s not a raven.
Above, however, we said that anything that is logically consistent with the hypothesis is evidence that the hypothesis is true. Finding an orange thing that is a traffic cone (and not a raven) is logically consistent with the hypothesis. So is the traffic cone evidence that all ravens are black? Logically speaking, yes, it is. As is every non-black, non-raven object in the universe. It turns out we’re surrounded at all times by overwhelming mountains of evidence that all ravens are black. The apple you had for lunch was evidence that all ravens are black. The Antikythera Mechanism in Athens is evidence that all ravens are black. The thousands of galaxies in the Virgo cluster are composed of quadrillions of solar masses of evidence that all ravens are black -- and they’re all so far away that a signal sent to them announcing the evolution of the corvid family wouldn’t even have reached the halfway mark yet. This is fairly counter-intuitive, to say the least.
There are several possible solutions to this conundrum. Hempel himself argued that the issue was in the assumptions we bring with us to the hypothesis. If you actually knew absolutely nothing about ravens and non-ravens, then you would more naturally consider every non-black, non-raven object as evidence of the hypothesis. Many other answers are focused on trying to parse out how much proof each piece of evidence provides. There are so many more non-raven objects in the universe than raven objects that every black raven provides astronomically more evidence for the hypothesis. This is a strong response, though in more complicated examples it gets hard to puzzle out exactly how much evidence something “should” be providing. But all the same, even the simplest assumptions about how scientific evidence works can quickly lead to weird conclusions.