proposition:  crime deserves punishment) is valid, even though no crime had ever been committed, and none ever punished.  Existence is not at all involved in universal propositions; “general knowledge lies only in our own thoughts, and consists barely in the contemplation of our own abstract ideas” and their relations.  The truths of mathematics and ethics are both universal and certain, while in natural science single observations and experiments are certain, but not general, and general propositions are only more or less probable.  Both the particular experiments and the general conclusions are of great value under certain circumstances, but they do not meet the requirements of comprehensive and certain knowledge.

The extent of our knowledge is very limited—­much less, in fact, than that of our ignorance.  For our knowledge reaches no further than our ideas, and the possibility of perceiving their agreements.  Many things exist of which we have no ideas—­chiefly because of the fewness of our senses and their lack of acuteness—­and just as many of which our ideas are only imperfect.  Moreover, we are often able neither to command the ideas which we really possess, or at least might attain, nor to perceive their connexions.  The ideas which are lacking, those which are undiscoverable, those which are not combined, are the causes of the narrow limits of human knowledge.

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.