I'm definitely open to critique here. I see ampliative reasoning (basically, induction or abduction) as different than explicative reasoning (deduction) in that ampliative reasoning adds additional information not contained in the premises. By some people's standards, this is never valid, and I suppose supporters of ampliative reasoning must show how this trick is possible.
The understanding that I've kind of developed is that ampliative reasoning adds information by decreasing likelihood. For instance, Pr(Socrates is a human mortal)=99% ⊢ Pr(All humans are mortal)=1%. What would otherwise be an audacious generalization is accounted for by decreasing the likelihood of the assertion.
My problem is that I don't even really know if this works out. The numbers in the Socrates example are completely made up, but my sense is that there are a set of probabilities that would make this inference valid. Peirce once wrote that with ampliative reasoning you trade in some security for uberty, and maybe I'm making too much of this relationship.
So my sense is that we're trying to quantify the information in both sentences, and for the inference to be valid, you can't add information to the inference, just like in deduction. But this means, in some sense, by saying that a proposition is merely probable, we say that the proposition contains less information.
For instance, "Socrates is a human mortal" contains a lot less information than "All human beings are mortal", because the latter tells us something about every human. But is there a way of quantifying the amount of information in these sentences?
If there is a way of quantifying the information in these two sentences, such that for instance the second sentence contains a thousand times the quantity of information as the first, then for the inference to be valid, would we just make the probability of the second sentence one-thousandth times the probability of the first?
What do you think?