Mr. John Stuart Mill, in his System of Logic, takes for granted that there _must_ be immediate, indemonstrable truths, to serve as a basis for deduction; "that there cannot be a chain of proof suspended from nothing;" that there must be ultimate laws of nature, though we cannot be sure that the laws now known to us are ultimate. On the other hand, we read in the recent work of an acute contemporary philosopher, Professor Delboeuf (Essai de Logique Scientifique, Liège, 1865, Pref. pp. v, vii, viii, pp. 46, 47:)--"Il est des points sur lesquels je crains de ne m'être pas expliqué assez nettement, entre autres la question du fondement de la certitude. Je suis de ceux qui repoussent de toutes leurs forces l'axiome si spécieux qu'on ne peut tout démontrer; cette proposition aurait, à mes yeux, plus besoin que toute autre d'une démonstration. Cette démonstration ne sera en partie donnée que quand on aura une bonne fois énuméré toutes les propositions indémontrables; et quand on aura bien défini le caractère auquel on les reconnait. Nulle part on ne trouve ni une semblable énumération, ni une semblable définition. On reste à cet égard dans une position vague, et par cela même facile à défendre." It would seem, by these words, that M. Delboeuf stands in the most direct opposition to Aristotle, who teaches us that the [Greek: a)rchai\] or _principia_ from which demonstration starts cannot be themselves demonstrated. But when we compare other passages of M. Delboeuf's work, we find that, in rejecting all undemonstrable propositions, what he really means is to reject all _self-evident universal truths_, "C'est donc une véritable illusion d'admettre des vérités évidentes par elles-mêmes. Il n'y a pas de proposition fausse que nous ne soyons disposés d'admettre comme axiome, quand rien ne nous a encore autorisés à la repousser" (p. ix.). This is quite true in my opinion; but the immediate indemonstrable truths for which Aristotle contends as [Greek: a)rchai\] of demonstration, are not announced by him as _self-evident_, they are declared to be results of sense and induction, to be raised from observation of particulars multiplied, compared, and permanently formularized under the intellectual _habitus_ called Noûs. By Demonstration Aristotle means deduction in its most perfect form, beginning from these [Greek: a)rchai\] which are inductively known but not demonstrable (_i. e._ not knowable deductively). And in this view the very able and instructive treatise of M. Delboeuf mainly coincides, assigning even greater preponderance to the inductive process, and approximating in this respect to the important improvements in logical theory advanced by Mr. John Stuart Mill.