This chapter examines two notable attempts made in recent times to justify inductive reasoning by means of a mathematical theory of probability, and argues that the both fail because they mistakenly reverse the logical relation between probability and induction. Attempts at justifying inductive argument are sometimes criticized for implicitly making the demand that it should lead to conclusions which are certain. The thesis to which the objection is made runs somewhat as follows: If an argument from true premises is valid its conclusion must be true, if true, then certain; so if induction is valid, since it's observed premises are true its generalized conclusion should be. Mathematical probability belongs to the latter type and is a calculation of chances based on the assumption that the alternatives are all equally possible. The empiricist predicament is plainly revealed in Professor Goodman's explanation of how what he calls the 'old problem' of induction has been 'solved or dissolved'.