It has been thought for many years (especially since Hume’s first Enquiry1) that induction lacks a justification. This justification for induction, it has been thought for long, is necessarily if it is rational to use induction. The argument for why there is no justification, indeed there cannot be a justification, can be presented like this:
(A1)
| n | Proposition | Explanation |
| 1 | If induction cannot be justified inductively or deductively, then it cannot be justified. | Premise |
| 2 | Induction cannot be justified inductively. | Premise |
| 3 | Induction cannot be justified deductively. | Premise |
| 4 | Induction cannot be justified. | From 1, 2, 3, Conj., MP. |
| 5 | Induction is not justified. | From 4, M2 |
It has then been thought that since we only ought to use justified reasoning systems and that induction is not a justified reasoning system, then we ought not to use induction, and thus that using induction was in some sense irrational.
But something is terribly wrong with this approach. First, is it really irrational to use induction? Is induction and deduction not what defines human rationality? What other reasoning systems are available to us? Second, an argument analogous to (A1) above can be constructed against deduction:
(A2)
| n | Proposition | Explanation |
| 1 | If deduction cannot be justified inductively or deductively, then it cannot be justified. | Premise |
| 2 | Deduction cannot be justified inductively. | Premise |
| 3 | Deduction cannot be justified deductively. | Premise |
| 4 | Deduction cannot be justified. | From 1, 2, 3, Conj., MP. |
| 5 | Deduction is not justified. | From 4, M |
Then, should we really accept that since deduction is not justified, then we ought not to use it? That’s a paradoxical conclusion since we just used deduction to reach it! Something is terribly wrong.
It seems a good idea to dispose with the premise used above that:
(P1). For all reasoning systems, if it is not justified, then we ought not to use it.
That seems to do the job. Now we can no longer infer that we ought not to use deduction or induction. But now the special problem for induction seems to have “vanished”, there is no problem for induction that there is not for deduction too. What is curious is that deduction has tended to be assumed to be a good reasoning system (and not in need of justification) while induction had to have a justification before we could rationally use it. Call adherents to this view deductivists.
But an inductivist, that is, someone who thought of it the other way could use (A2) above against the deductivist.3
1An Enquiry concerning Human Understanding published 1748.
2M is an alethic logic axiom. It follows from the definition of ¬◊, substitution and M.
3Note that the argument will not have any force for the inductivist since he does not trust deduction but the deductivist does.