So, we've been talking about this in Philosophy, and I wanted to know what you all thought.
Basically, it starts with a few basic principles:
G1: Generalizations are confirmed by their instances. That is, if you have a hypothesis, any result that lends support to that thesis will confirm said theory--which does not mean it's irrefutable, just that it has some degree of confirmation.
E1: If two hypotheses are known to be logically equivalent, then any data that supports 1 also supports the other. This one kind of explains itself, and is easily seen.
Now, some generalizations:
R1: All Ravens are black things.
R2: All non-black things are non-Ravens.
L1: R1 and R2 are logically equivalent.
Now, then. That's all we need to lay out the paradox. And here it comes:
A sneaker is a non-black non-raven, therefore it lends support to R1; that all Ravens are black things.
Agree? Disagree?
Some proposed solutions/objections:
Solution 1: The conclusion the paradox comes up with is correct; we only find it so infeasible because we assume the degrees of support lent to be equal; that is, for one non-black thing to lend as much support as a black raven. However, they actually lend support equal to 1/(total number of objects). So one black raven out of, say, 6 million in the world would lend support equal to 1/6,000,000. However, there are trillions upon trillions upon trillions of non-black non-raven things. So the support one would lend to R1 would be small: on the order of 1/googol^(googol^(googol)), which is significantly less. It does lend support, but not nearly as much.
Objection 1: Not all generalizations are confirmed by their instances. For instance, take the theory that "All snakes are non-Irish." You go out, and start finding non-Irish snakes. First in proximity, but then further and further away. The more you find, and in the greater variety of ecosystems you find them (say you found them in, say, Scotland), the more likely you are to believe that they also dwell in Ireland. Therefore, this disproves the idea that all generalizations are confirmed by their instances.
Thoughts?
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4 comments:
I would agree with what solution 1 says; finding non-black non-raven things would technically make it more likely that both hypotheses are true, but in reality the evidence would be so small that it would be meaningless because there are so many things. If the hypothesis were set in a closed system with a limited population of ravens and other non-raven things then finding all or most of the non-black things and seeing that they are not ravens would certainly help prove or support the hypothesis.
I don’t really agree with the objection there. The generalization that “all snakes are non-Irish” would indeed be supported by instances like the ones used in the raven problem; every non-Irish snake found would lend support to that theory, and finding one Irish snake would disprove it. The existence of snakes in nearby or similar regions would make it seem likely that there were also snakes in Ireland, but that’s a more abstract way of supporting the hypothesis and it doesn’t really have any relevance to the question about the ravens.
i got halfway through and got confused and gave up. i have no will power
Well, John, I'm inclined to agree with you. I think the objection is kind of similar to an assertion like: because we find so many non-black non-ravens of so many colors, we are more inclined to believe there is a non-black raven. Or that because we find so many red things, for instance, we're more inclined to believe there is a red raven--but that just doesn't hold when applied to this theory. Though that's kind of pigeon-holing the objection that it's bringing up, which is merely a rejection of G1.
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