In case you have not seen this already, I have written a piece on Hume and the problem of induction, or to be more precise and less general, the problems of inductivism:
Hopefully—space allows-an extended version of this piece covering more ground regarding Putnam’s argument against the possibility of a Universal or optimal learning machine, and Solomonoff’s formal account of Occam’s razor.
Nevertheless since writing this text, I have come to the conclusion that there are some problems particularly with regard to precision. For example, a less serious issue is my rather dodgy treatment of Carnap’s view on a formal learning machine in his magnum opus Logical Foundations of Probability. I have gone along with Putnam’s argument but the issue is that Putnam’s challenge in Inductive Logic and Degree of Confirmation, and in his address to Radio Free Europe (Probability and Confirmation) are not false. They attribute a view to Carnap which is not accurate. In other words, Putnam takes the scope of a formal learning machine—one that Carnap mentions toward the end of his book—as far broader and more ambitious than what Carnap thinks to be the case.
A more serious problem is the one recently mentioned to me by my friend Adam Berg: That Goodman’s new riddle (the problem of projectibles) and Putnam’s take on the problem of induction differ on a fundamental ground and cannot be treated as if they were both tackling the same problem of induction. In one case, the problem is explicitly dealing with observations or empirical statements, whereas in the other case—i.e. Carnap’s inductive logic which is the target of Putnam’s critique—such observations are absent.
In the case of the latter, we do not have simple observational statements. All we have are logical statements. Even the e-statement in c(h, e)=r is only a reference—within the framework of an inductive logic—and not an empirical observation per se. To this extent, one should be cautious to use examples like Raven or Grue paradox (i.e. explicitly observation-based inductive paradoxes) to challenge Carnap’s inductive logic. As a resolution and a mediation between the two views, Adam has asked me to look into Reichenbach’s paradigm of induction as vindication. In order to that, I need to revisit Reichenbach’s texts on this subject. Yet I think there is still a shadow that haunts even Reichenbach’s paradigm of induction whose flaws are spectacularly—albeit inadvertently—highlighted in Sellars’s essay Induction as Vindication. This shadow is the problem of simplicity or elegance. More accurately, it is the problem of an unconstrained account of simplicity whose espousal demands a metaphysical high-price (e.g., see Grünbaum or Rescher’s critiques). Even the formal conception of simplicity has its own incoherencies which are addressed in my piece.