Sam asked:

What is the Humean problem of induction? If generalizations cannot be rationally grounded, then why do we believe in them?

Answer by Helier Robinson

The problem of induction is the problem that induction is logically invalid. Induction is arguing from some cases to all cases; in science it is argument from all cases so far observed to all unobserved cases. For example, energy has so far always been observed to be conserved, so it is argued that energy is always conserved. In daily life induction is generalisation, and its unreliability is shown by the falsity of superstition and stereotyping. The general case may be put in the form of sets: if a set P is a subset of another set Q, and it is known that something is true of every member of P, is it also true of every member of Q? The answer is, of course, maybe and maybe not.

There is an exception to this, involving intensions. The intension of a set, if it has one, is that property or properties that all and only the members of the set have; that is, every member of the set has that property or properties, and everything that has that property or properties is a member of that set. If now Q has an intension, and what is true of every member of P is true because of this intension, then it is true of every member of Q.

This is significant in philosophy of science. There are two branches of science, empirical and theoretical. Empirical science collects data, in the observatory, field, or laboratory, finds patterns among the data, and generalises these patterns into scientific laws. Theoretical science tries to explain these laws by describing their underlying causes. Explanation is causal: to describe causes is to explain their effects; but causes are never perceived (only correlations are perceived) and therefore are not empirical, so are theoretical. Empirical science is not intensional, so has the problem of induction, while theoretical science is intensional and so does not have that problem (That is, empirical science generalises patterns into laws, and these generalisations are inductive.) On the other hand, explanations — hence theories — can be wild and wooly, as some old fashioned metaphysics used to be. Science gets round these two difficulties by relying on accurate observation for truth, and by requiring theories to correspond to empirical fact. For example, modern theoretical physics requires the principle of conservation of energy to be true by Noether’s theorem.