It is plain from Aristotle’s own words[64] that he intended these four chapters (xix.-xxii.) as a confirmation of what he had already asserted in chapter iii. of the present treatise, and as farther refutation of the two distinct classes of opponents there indicated: (1) those who said that everything was demonstrable, demonstration in a circle being admissible; (2) those who said that nothing was demonstrable, inasmuch as the train of predication upwards, downwards, and intermediate, was infinite. Both these two classes of opponents agreed in saying, that there were no truths immediate and indemonstrable; and it is upon this point that Aristotle here takes issue with them, seeking to prove that there are and must be such truths. But I cannot think the proof satisfactory; nor has it appeared so to able commentators either of ancient or modern times — from Alexander of Aphrodisias down to Mr. Poste.[65] The elaborate amplification added in these last chapters adds no force to the statement already given at the earlier stage; and it is in one respect a change for the worse, inasmuch as it does not advert to the important distinction announced in chapter iii., between universal truths known by Induction (from sense and particulars), and universal truths known by Deduction from these. The truths immediate and indemonstrable (not known through a middle term) are the inductive truths, as Aristotle declares in many places, and most emphatically at the close of the Analytica Posteriora. But in these chapters, he hardly alludes to Induction. Moreover, while trying to prove that there must be immediate universal truths, he neither gives any complete list of them, nor assigns any positive characteristic whereby to identify them. Opponents might ask him whether these immediate universal truths were not ready-made inspirations of the mind; and if so, what better authority they had than the Platonic Ideas, which are contemptuously dismissed.
[64] Analyt. Post. I. xxii. p. 84, a. 32: ὅπερ ἔφαμέν τινας λέγειν κατ’ ἀρχάς, &c.
[65] See Mr. Poste’s note, p. 77, of his translation of this treatise. After saying that the first of Aristotle’s dialectical proofs is faulty, and that the second is a petitio principii, Mr. Poste adds, respecting the so-called analytical proof given by Aristotle:— “It is not so much a proof, as a more accurate determination of the principle to be postulated. This postulate, the existence of first principles, as concerning the constitution of the world, appears to belong properly to Metaphysics, and is merely borrowed by Logic. See Metaph. ii. 2, and Introduction.â€� In the passage of the Metaphysica (α. p. 994) here cited the main argument of Aristotle is open to the same objection of petitio principii which Mr. Poste urges against Aristotle’s second dialectical argument in this place.
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 Delbœuf (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. DelbÅ“uf stands in the most direct opposition to Aristotle, who teaches us that the ἀρχαὶ or principia from which demonstration starts cannot be themselves demonstrated. But when we compare other passages of M. DelbÅ“uf’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 repousserâ€� (p. ix.). This is quite true in my opinion; but the immediate indemonstrable truths for which Aristotle contends as ἀρχαὶ 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 ἀρχαὶ which are inductively known but not demonstrable (i. e. not knowable deductively). And in this view the very able and instructive treatise of M. DelbÅ“uf 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.
Among the universal propositions which are not derived from Induction, but which serve as ἀρχαὶ for Deduction and Demonstration, we may reckon the religious, ethical, æsthetical, social, political, &c., beliefs received in each different community, and impressed upon all newcomers born into it by the force of precept, example, authority. Here the major premiss is felt by each individual as carrying an authority of its own, stamped and enforced by the sanction of society, and by the disgrace or other penalties in store for those who disobey it. It is ready to be interpreted and diversified by suitable minor premisses in all inferential applications. But these ἀρχαὶ for deduction, differing widely at different times and places, though generated in the same manner and enforced by the same sanction, would belong more properly to the class which Aristotle terms τὰ ἔνδοξα.
We have thus recognized that there exist immediate (ultimate or primary) propositions, wherein the conjunction between predicate and subject is such that no intermediate term can be assigned between them. When A is predicated both of B and C, this may perhaps be in consequence of some common property possessed by B and C, and such common property will form a middle term. For example, equality of angles to two right angles belongs both to an isosceles and to a scalene triangle, and it belongs to them by reason of their common property — triangular figure; which last is thus the middle term. But this need not be always the case.[66] It is possible that the two propositions — A predicated of B, A predicated of C — may both of them be immediate propositions; and that there may be no community of nature between B and C. Whenever a middle term can be found, demonstration is possible; but where no middle term can be found, demonstration is impossible. The proposition, whether affirmative or negative, is then an immediate or indivisible one. Such propositions, and the terms of which they are composed, are the ultimate elements or principia of Demonstration. Predicate and subject are brought constantly into closer and closer conjunction, until at last they become one and indivisible.[67] Here we reach the unit or element of the syllogizing process. In all scientific calculations there is assumed an unit to start from, though in each branch of science it is a different unit; e.g. in barology, the pound-weight; in harmonics, the quarter-tone; in other branches of science, other units.[68] Analytical research teaches us that the corresponding unit in Syllogism is the affirmative or negative proposition which is primary, immediate, indivisible. In Demonstration and Science it is the Noûs or Intellect.[69]
[66] Analyt. Post. I. xxiii. p. 84, b. 3-18. τοῦτο δ’ οὐκ ἀεὶ οὕτως ἔχει.
[67] Ibid. b. 25-37. ἀεὶ τὸ μέσον πυκνοῦται, ἕως ἀδιαίρετα γένηται καὶ ἕν. ἔστι δ’ ἕν, ὅταν ἄμεσον γένηται καὶ μία πρότασις ἁπλῶς ἡ ἄμεσος.