Recently I tried to explain to a friend what interested me about Hume's 'problem of induction.' I told him how if we want to give an argument for the superiority of inductive reasoning (concluding x's are always P, based on observed instances of x's that are P) over, say, anti-inductive reasoning (concluding x's are not always P, based on observed instances of x's that are P) then we would have to give either an inductive argument or else a deductive argument. We cannot give such a deductive argument, I told him, and to give an inductive argument like 'inductive reasoning has led to good results in every observed instance' would be circular.
He replied with the question 'why is there no problem of deduction?' He asked why he couldn't give a similar argument that any defense of deductive reasoning (concluding C based on premises that logically entail C) over, say, anti-deductive reasoning (concluding not C based on premises that logically entail C) needs to be either deductive or inductive. A deductive argument would be circular, and an inductive argument is inadequate because of Hume's problem.
I can't shake the feeling that something is wrong with his reply. Is there - or is there something wrong in both of our arguments? If not, then why is 'the problem of induction' so much more famous than 'the problem of deduction'?