Besides real existence Locke, unsystematically enough, enumerates three other sorts of agreement between ideas,—­in the perception of which he makes knowledge consist,—­viz., identity or diversity (blue is not yellow), relation (when equals are added to equals the results are equal), and coexistence or necessary connexion (gold is fixed).  We are best off in regard to the knowledge of the first of these, “identity or diversity,” for here our intuition extends as far as our ideas, since we recognize every idea, as soon as it arises, as identical with itself and different from others.  We are worst off in regard to “necessary connexion.”  We know something, indeed, concerning the incompatibility or coexistence of certain properties (e. g., that the same object cannot have two different sizes or colors at the same time; that figure cannot exist apart from extension):  but it is only in regard to a few qualities and powers of bodies that we are able to discover dependence and necessary connexion by intuitive or demonstrative thought, while in most cases we are dependent on experience, which gives us information concerning particular cases only, and affords no guarantee that things are the same beyond the sphere of our observation and experiment.  Since empirical inquiry furnishes no certain and universal knowledge, and since the assumption that like bodies will in the same circumstances have like effects is only a conjecture from analogy, natural science in the strict sense does not exist.  Both mathematics and ethics, however, belong in the sphere of the demonstrative knowledge of relations.  The principles of ethics are as capable of exact demonstration as those of arithmetic and geometry, although their underlying ideas are more complex, more involved, hence more exposed to misunderstanding, and lacking in visible symbols; though these defects can, and should, in part be made good by careful and strictly consistent definitions.  Such moral principles as “where there is no property there is no injustice,” or “no government allows absolute liberty,” are as certain as any proposition in Euclid.

The advantage of the mathematical and moral sciences over the physical sciences consists in the fact that, in the former, the real and nominal essences of their objects coincide, while in the latter they do not; and, further, that the real essences of substances are beyond our knowledge.  The true inner constitution of bodies, the root whence all their qualities, and the coexistence of these, necessarily proceed, is completely unknown to us; so that we are unable to deduce them from it.  Mathematical and moral ideas, on the other hand, and their relations, are entirely accessible, for they are the products of our own voluntary operations.  They are not copied from things, but are archetypal for reality and need no confirmation from experience.  The connexion constituted by our understanding between the ideas crime and punishment (e. g., the