Amusing note on induction.

Russell was very concerned with the problem of induction. According to Hume, the principle of induction is synthetic, and therefore, any attempt to justify it inductively would beg the question. Must the principle of induction be presupposed without a reason? If this is a problem, then induction is peculiar indeed.

Take the following inductive inference:

a is *y*, b is *y*, c is *y* |= every *x* is *y*

This inference can be improved, or made more "cogent" by adding premises, for example:

a is *y*, b is *y*, c is *y*, d is *y* |= every *x* is *y*

Again:

a is *y*, b is *y*, c is *y*, d is *y*, e is *y* |= every *x* is *y*

And so on.

So what is the perfect inductive inference? Which inductive inference would be the most cogent? Well, presumably it is the argument with infinitely many premises:

a is *y*, b is *y*, c is *y*, d is *y*, e is *y*, ... |= every *x* is *y*

But 'every *x* is *y*' is now equivalent to the premises, and this is no inductive inference, but deductive! Moreover, the premises and conclusion are equal, that is, they say the same thing by different methods. In other words, the inference does not just beg the question, but is circular! Apparently, inductive inferences become stronger as more and more of the conclusion is begged. That is, the more closely an inductive inference resembles a circular inference the better, and yet despite striving to beg the question, Russell thought it a problem, too.