Once More on Black Ravens
In spite of the fact that I consider my two previous posts on the "Raven paradox" to be correct as far as they go, I have still been drawn to this problem because I have not addressed a central issue: namely, how can enumerating non-black objects possibly prove that all ravens are black?
To restate the issue, stating that "all ravens are black" is supposed to be logically equivalent to "all non-black objects are not ravens." Therefore, if I find a non-black object and ascertain that it is not a raven, I have, theoretically at least, contributed to proving that all ravens are black.
This seems counterintuitive, since I haven't even seen a raven and yet I have somehow helped to prove that all ravens are black. Let's reduce the problem to marbles. There is a bin filled with marbles, some of which are translucent, which we will identify with "ravens," and some are opaque, which correspond to "not ravens." There are 100 marbles in the bin, some of which are ravens and some not. We pull out a marble, discover that it is white and opaque. We have therefore found a non-black non-raven, and hence have contributed to prove that all ravens are black.
There is a logical leap, I believe in saying that "all ravens are black" and "all non-black objects are not ravens" are equivalent statements, and saying that evidence for each helps prove the other. If all the marbles in the bin were visible to us, we could easily separate out the black marbles from the non-black, and then search through the non-black ones to see if we find any ravens. If we don't, no one would dispute that we have proved that all ravens are black. Drawing a random marble out of the bin, however, is less convincing. There is at least a possibility, as far as we know, that we could have drawn out a non-black raven, and that would disprove our hypothesis. (I am overlooking my objections to treating material objects as Platonic categories, which we have effectively done by creating an artificial environment of discrete objects with discrete characteristics (black or non-black, raven or non-raven). I am interested purely in the logical side of the issue in this case.) Therefore, every time we don't pull out a non-black raven, we have contributed probabilistically to demonstrate the correctness of our theorem. Hempel's solution to the paradox is that we indeed contribute to prove the theorem, although in the real world there is a virtual infinity of non-black objects and rather fewer ravens, so going about it by finding non-black objects that aren't ravens is an infinitesimally small contribution. Theoreticallyl, if we could view all the objects in the world and separate the non-black ones from the black ones, and then sort through the non-black ones to see if there are any ravens, this would still prove our hypothesis, just as in the marble simulation described above.
Okay, but philosophers have objected that we are somehow contributing to prove that ravens are black when we identify non-black objects in our houses. How can I notice that a piece of paper is white and not a raven, and have that contribute to a proof that all ravens are black?
Let's think of another situation with marbles. We have one bin that consists of ravens and non-ravens, as before, and a second bin that consists of only non-ravens. If I pull a marble out of the bin that consists of non-ravens, does anyone believe that I have contributed to prove anything about ravens? No, because we knew in advance that there would not be a raven in there. If I took the entire bin and separated it out into black and non-black objects, as before, and then noted that none of the non-black marbles were ravens, I have not contributed to prove that all ravens are black, because we knew in advance that there would be no ravens among the marbles. It is only when ravens are among the objects that we are searching that we can reasonably assert that finding a non-black non-raven is really helpful toward proving that all ravens are black. This means that, unless I have a menagerie that may include ravens in my house, I could list the colour of every single object and I would not contribute anything toward proving that ravens are black. In fact, if I go out in the field to look for ravens and I find leaves and water and insects that are not black, I would also be contributing nothing to prove that ravens are not black, because I am essentially pulling marbles from the bin of non-ravens.
It would be different if I were a robot that could not distinguish anything without examining it closely. In that case, even inside a house, I might have no knowledge of ravens' habits, and indeed if I knew nothing about the owners of the houses even a human would not know in advance that there are no ravens within. The robot must choose an object, examine it for colour and type at the same time, and then declare if it has found a non-black non-raven. If such a robot could cover every object on earth, he could tell us definitively if there are any non-black ravens, and therefore prove the hypothesis; and, even without covering every object, we could assume that it had investigated a random sample and therefore that a number of non-black non-ravens would contribute to prove that all ravens are black (in however small a way). Not so if humans put the robot in a particular area and set it to investigate. If we set it in our own house and told it to come up with a list of all non-black objects, we would be looking in the bin of non-ravens and therefore we would be contributing nothing to our hypothesis.
To restate the issue, stating that "all ravens are black" is supposed to be logically equivalent to "all non-black objects are not ravens." Therefore, if I find a non-black object and ascertain that it is not a raven, I have, theoretically at least, contributed to proving that all ravens are black.
This seems counterintuitive, since I haven't even seen a raven and yet I have somehow helped to prove that all ravens are black. Let's reduce the problem to marbles. There is a bin filled with marbles, some of which are translucent, which we will identify with "ravens," and some are opaque, which correspond to "not ravens." There are 100 marbles in the bin, some of which are ravens and some not. We pull out a marble, discover that it is white and opaque. We have therefore found a non-black non-raven, and hence have contributed to prove that all ravens are black.
There is a logical leap, I believe in saying that "all ravens are black" and "all non-black objects are not ravens" are equivalent statements, and saying that evidence for each helps prove the other. If all the marbles in the bin were visible to us, we could easily separate out the black marbles from the non-black, and then search through the non-black ones to see if we find any ravens. If we don't, no one would dispute that we have proved that all ravens are black. Drawing a random marble out of the bin, however, is less convincing. There is at least a possibility, as far as we know, that we could have drawn out a non-black raven, and that would disprove our hypothesis. (I am overlooking my objections to treating material objects as Platonic categories, which we have effectively done by creating an artificial environment of discrete objects with discrete characteristics (black or non-black, raven or non-raven). I am interested purely in the logical side of the issue in this case.) Therefore, every time we don't pull out a non-black raven, we have contributed probabilistically to demonstrate the correctness of our theorem. Hempel's solution to the paradox is that we indeed contribute to prove the theorem, although in the real world there is a virtual infinity of non-black objects and rather fewer ravens, so going about it by finding non-black objects that aren't ravens is an infinitesimally small contribution. Theoreticallyl, if we could view all the objects in the world and separate the non-black ones from the black ones, and then sort through the non-black ones to see if there are any ravens, this would still prove our hypothesis, just as in the marble simulation described above.
Okay, but philosophers have objected that we are somehow contributing to prove that ravens are black when we identify non-black objects in our houses. How can I notice that a piece of paper is white and not a raven, and have that contribute to a proof that all ravens are black?
Let's think of another situation with marbles. We have one bin that consists of ravens and non-ravens, as before, and a second bin that consists of only non-ravens. If I pull a marble out of the bin that consists of non-ravens, does anyone believe that I have contributed to prove anything about ravens? No, because we knew in advance that there would not be a raven in there. If I took the entire bin and separated it out into black and non-black objects, as before, and then noted that none of the non-black marbles were ravens, I have not contributed to prove that all ravens are black, because we knew in advance that there would be no ravens among the marbles. It is only when ravens are among the objects that we are searching that we can reasonably assert that finding a non-black non-raven is really helpful toward proving that all ravens are black. This means that, unless I have a menagerie that may include ravens in my house, I could list the colour of every single object and I would not contribute anything toward proving that ravens are black. In fact, if I go out in the field to look for ravens and I find leaves and water and insects that are not black, I would also be contributing nothing to prove that ravens are not black, because I am essentially pulling marbles from the bin of non-ravens.
It would be different if I were a robot that could not distinguish anything without examining it closely. In that case, even inside a house, I might have no knowledge of ravens' habits, and indeed if I knew nothing about the owners of the houses even a human would not know in advance that there are no ravens within. The robot must choose an object, examine it for colour and type at the same time, and then declare if it has found a non-black non-raven. If such a robot could cover every object on earth, he could tell us definitively if there are any non-black ravens, and therefore prove the hypothesis; and, even without covering every object, we could assume that it had investigated a random sample and therefore that a number of non-black non-ravens would contribute to prove that all ravens are black (in however small a way). Not so if humans put the robot in a particular area and set it to investigate. If we set it in our own house and told it to come up with a list of all non-black objects, we would be looking in the bin of non-ravens and therefore we would be contributing nothing to our hypothesis.
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