In the book 'Introduction of the theory of Statistics' by Mood,Graybill,Boes (third edition)on page 220 (Chapter 6 on Sampling) you can read: 'Inductive inference is well known to be a hazardous process.In fact,it is a theorem of logic that in inductive inference uncertainty is present.One simply cannot make absolutely certain generalization.' What theorem of logic do they refer to? Can you give me please some reference to this fundamental result ?
This is an elementary but I think reasonable explanation of the difference between deductive and inductive reasoning. The authors make clear that most arguments can be framed either way, but require different types of support. In their example, one can argue that a kicked ball will fall to the ground by appeal to Newton's law (deductive) or by reference to previous instances in which balls have fallen (inductive).
This highlights the risk of induction. Absent a general rule, inference (induction) can lead one in the wrong direction. We would not want to say that a fair roulette wheel will land on black simply because it has done so several times in a row.
Hope this helps.
The use of the term "theorem" seems rather strange. It at least doesn't feel like we can formalize this idea quite in the same way logical theorems, and mathematical theorems, can get formalized.
That said, it isn't hard to make the case that uncertainty has to come as present in inductive inference. Let us suppose that uncertainty does not exist in inductive inference (this presupposes that uncertainty either exists or does not exist in a given situation, which seems safe here). It then follows that inductive inference would qualify as certain. So, inductive inference would have to always work. But, it doesn't always work, as plenty of examples can get supplied where it leads us in the wrong direction. So, inductive inference doesn't qualify as certain, and consequently in inductive inference uncertainty is present.
"Truism of logic" or "truth of logic" might work as better here than "theorem of logic". I doubt that by "theorem" the authors meant anything more than "truth of logic", though I don't know for certain.