There are two ways by which knowledge may be extended: by experience, on the one hand, and, on the other, by the elevation of our ideas to a state of clearness and distinctness, together with the discovery and systematic arrangement of those intermediate ideas which exhibit the relation of other ideas, in themselves not immediately comparable. The syllogism, as an artificial form, is of little value in the perception of the agreements between these intermediate and final terms, and of none whatever in the discovery of the former. Analytical and identical propositions which merely explicate the conception of the subject, but express nothing not already known, are, in spite of their indefeasible certitude, valueless for the extension of knowledge, and when taken for more than verbal explanations, mere absurdities. Even those most general propositions, those "principles" which are so much talked of in the schools, lack the utility which is so commonly ascribed to them. Maxims are, it is true, fit instruments for the communication of knowledge already acquired, and in learned disputations may perform indispensable service in silencing opponents, or in bringing the dispute to a conclusion; but they are of little or no use in the discovery of new truth. It is a mistake to believe that special cases (as 5 = 2 + 3, or 5 = 1 + 4) are dependent on the truth of the abstract rule (the whole is equal to the sum of its parts), that they are confirmed by it and must be derived from it. The particular and concrete is not only as clear and certain as the general maxim, but better known than this, as well as earlier and more easily perceived. Nay, further, in cases where ideas are confused and the meanings of words doubtful, the use of axioms is dangerous, since they may easily lend the appearance of proved truth to assertions which are really contradictory.