I may briefly indicate here how I think the problem ought to be attacked. “Objective facts” can only be established by induction. I do not mean by that term necessarily the process described by Mill, but some similar process, based ultimately on _inductio per enumerationem simplicem_. Now no such process can ever lead to a necessary truth. The most fundamental and certain induction which can be made, that which induces us to believe in the objectivity of our environment, does not lead to a “necessary truth”; and much less can any other induction based upon this one do so. “Objective facts” then may be established with greater or less probability, but can never be _necessarily_ true. But all inductions are based on our perceptions, that is ultimately on our subjective sensations. And a man can, nay, must be, absolutely certain of the reality of his own sensations though he cannot be certain of the interpretations he puts upon them. If I have a toothache I cannot be absolutely certain that I have a tooth, but, at least while the pain lasts, I am _absolutely_ certain that I have an ache. And so of any subjective sensation. I can similarly be absolutely certain that I entertain a given concept, while that concept is before my mind; though of course it is possible that if I assert the possession of that concept I may do so in language which may be misunderstood by the person I am addressing. If then a man has certain concepts which he can call up at will, the reality of those concepts, _qua_ concepts, is to him a _necessary truth_. He may lay down such necessary truths as axioms, and by their aid he may give real subjective import to a symbolic argument, and so obtain new and complicated assertions which are also to him necessary truths. This is what I do in my subjective theory of geometry. That theory might be regarded as purely symbolic—the axioms might have been left out, and all its conclusions looked upon as mere truisms. The conclusions of geometry of four or more independent directions can perhaps _only_ be regarded as truisms. But by the aid of the axioms, geometry of two and three independent directions can be given real subjective import, and its conclusions therefore regarded as necessary truths, as long as they are only taken subjectively. They may further be applied objectively by the aid of objective facts established by induction, but in this case their validity is no greater than that of the primary facts, the counterparts of the subjective axioms, which are employed to give the theory objective import.