Inductive knowledge

The inductive method suffered a devastating criticism by David Hume (Hume, An inquiry concerning human understanding, 1748),

"All inferences from experience, therefore, are effects of custom, not of reasoning"

whose consequences extend up-to the twentieth century. In fact, Bertrand Russell argues that Hume "represents the bankruptcy of rationality" (A history of western philosophy, 1946). 

Later, the philosopher Karl Popper tried  to justify inductive methods from logic. According to Popper (Objective Knowledge, 1972), it is possible to prefer one hypothesis to another based on empirical justifications: since the experimental facts can refute some hypothesis, one should prefer one whose falsity has not been proved by facts. 

From this point of view, there is however no reason for us to choose any particular one among the many hypothesis which have not been refuted by the experience: the fact that a hypothesis matches all the experiments, it does not matter how large the amount of them, does not guarantee that the

agreement will remain true in future experiments. In other words, the experiments cannot verify a hypothesis, they only refute some of them.

To solve this concern, Popper introduced the concept of simplicity in the induction process:

"We have to value more the simple statements than those not so simple, because they tell us more, because their empirical content is larger and they can be validated more easily".

Promising as it is at first glance, this definition of simplicity (based on the degree of testability) is however unpractical, and lacks precision in order to be applied in general (see, for instance, Carl G. Hempel's criticisms in Philosophy of Natural Science, 1966) .

The concept of identification in the limit (E. M. Gold, Language identification in the limit,  Information and Control 10 (447–474), 1967) provides a rational criterion to choose a hypothesis: under certain circumstances, some procedures (but not all) guarantee that the correct hypothesis will be the one selected after a large enough set of observations (although it may not be possible to specify what is meant in a particular case by "large enough").

Indeed, for any method used to sort (a numerable set of) assumptions, the criterion "select the first hypothesis which agrees with all the observations" allows for the identification of the correct hypothesis in the limit. This theoretical result suggests a rigorous (although flexible) definition for the  concept of simplicity and provides a logical support for the inductive approach.