Compare Metaph. K. vii. p. 1063, b. 36; p. 1065, a. 8-26. Analyt. Post. I. ii. p. 71, b. 9.
There remain, as matter proper for the investigation of First Philosophy, the two last-mentioned heads of Ens; viz., Ens according to the Ten Categories, and Ens potential and actual. But, along with these, Aristotle includes another matter also; viz., the critical examination of the Axioms and highest generalities of syllogistic proof or Demonstration. He announces as the first principle of these Axioms — as the highest and firmest of all Principles — the Maxim of Contradiction:[12] The same predicate cannot both belong and not belong to the same subject, at the same time and in the same sense; or, You cannot both truly affirm, and truly deny, the same predicate respecting the same subject; or, The same proposition cannot be at once true and false. This Axiom is by nature the beginning or source of all the other Axioms. It stands first in the order of knowledge; and it neither rests upon nor involves any hypothesis.[13]
[12] Metaph. Γ. iii. p. 1005, b. 7, 17, 22, 34: αὕτη δὴ πασῶν ἐστὶ βεβαιοτάτη τῶν ἀρχῶν — φύσει γὰρ ἀρχὴ καὶ τῶν ἄλλων ἀξιωμάτων αὕτη πάντων. — p. 1011, b. 13: βεβαιοτάτη δόξα πασῶν τὸ μὴ εἶναι ἀληθεῖς ἅμα τὰς ἀντικειμένας φάσεις — (He here applies the term δόξα to designate this fundamental maxim. This deserves notice, because of the antithesis, common with him elsewhere, between δόξα and ἐπιστήμη).
[13] Metaph. Γ. iii. p. 1005, b. 13-14: γνωριμωτάτην — ἀνυπόθετον.
The Syllogism is defined by Aristotle as consisting of premisses and a conclusion: if the two propositions called premisses be granted as true, a third as conclusion must for that reason be granted as true also.[14] The truth of the conclusion is affirmed conditionally on the truth of the premisses; and the rules of Syllogism set out those combinations of propositions in which such affirmation may be made legitimately. The rules of the Syllogism being thus the rules for such conditional affirmation, the Principle or Axiom thereof enunciates in the most general terms what is implied in all those rules, as essential to their validity. And, since the syllogistic or deductive process is applicable without exception to every variety of the Scibile, Aristotle considers the Axioms or Principles thereof to come under the investigation of Ontology or First Philosophy. Thus it is, that he introduces us to the Maxim of Contradiction, and its supplement or correlative, the Maxim of the Excluded Middle.
[14] Analyt. Prior. I. i. p. 24, b. 18-20, et alib.
His vindication of these Axioms is very illustrative of the philosophy of his day. It cannot be too often impressed that he was the first either to formulate the precepts; or to ascend to the theory, of deductive reasoning; that he was the first to mark by appropriate terms the most important logical distinctions and characteristic attributes of propositions; that before his time, there was abundance of acute dialectic, but no attempt to set forth any critical scheme whereby the conclusions of such dialectic might be tested. Anterior to Sokrates, the cast of Grecian philosophy had been altogether either theological, or poetical, or physical, or at least some fusion of these three varieties into one. Sokrates was the first who broke ground for Logic — for testing the difference between good and bad ratiocination. He did this by enquiry as to the definition of general terms,[15] and by dialectical exposure of the ignorance generally prevalent among those who familiarly used them. Plato in his Sokratic dialogues followed in the same negative track; opening up many instructive points of view respecting the erroneous tendencies by which reasoners were misled, but not attempting any positive systematic analysis, nor propounding any intelligible scheme of his own for correction or avoidance of the like. If Sokrates and Plato, both of them active in exposing ratiocinative error and confusion, stopped short of any wide logical theory, still less were the physical philosophers likely to supply that deficiency. Aristotle tells us that several of them controverted the Maxim of Contradiction.[16] Herakleitus and his followers maintained the negative of it, distinctly and emphatically;[17] while the disciples of Parmenides, though less pronounced in their negative, could not have admitted it as universally true. Even Plato must be reckoned among those who, probably without having clearly stated to himself the Maxim in its universal terms, declared doctrines quite incompatible with it: the Platonic Parmenides affords a conspicuous example of contradictory conclusions deduced by elaborate reasoning and declared to be both of them firmly established.[18] Moreover, in the Sophistes,[19] Plato explains the negative proposition as expressing what is different from that which is denied, but nothing beyond; an explanation which, if admitted, would set aside the Maxim of Contradiction as invalid.
[15] Aristot. Metaph. A. vi. p. 987, b. 1: Σωκράτους δὲ περὶ μὲν τὰ ἠθικὰ πραγματευομένου, περὶ δὲ τῆς ὅλης φύσεως οὐθέν, ἐν μέντοι τούτοις τὸ καθόλου ζητοῦντος, καὶ περὶ ὁρισμῶν ἐπιστήσαντος πρώτου τὴν διάνοιαν.
[16] Aristot. Metaph. Γ. iv. p. 1005, b. 35: εἰσὶ δέ τινες, οἵ, καθάπερ εἴπομεν, αὐτοί τε ἐνδέχεσθαί φασι τὸ αὐτὸ εἶναι καὶ μὴ εἶναι, καὶ ὑπολαμβάνειν οὕτως. χρῶνται δὲ τῷ λόγῳ τούτῳ πολλοὶ καὶ τῶν περὶ φύσεως.
[17] Ibid. iii. p. 1005, b. 25; v. p. 1010, a. 13; vi. p. 1011, a. 24.