  To effect this sort of reduction, Aristotle relies upon a series of meta-theorems, some of which he proves and others of which he merely reports (though it turns out that they do all indeed admit of proofs). His principles are meta-theorems in the sense that no argument can run afoul of them and still qualify as a genuine deduction. They include such theorems as: (i) no deduction contains two negative premises; (ii) a deduction with a negative conclusion must have a negative premise; (iii) a deduction with a universal conclusion requires two universal premises; and (iv) a deduction with a negative conclusion requires exactly one negative premise. He does, in fact, offer proofs for the most significant of his meta-theorems, so that we can be assured that all deductions in his system are valid, even when their validity is difficult to grasp immediately.