The two Axioms announced in the Metaphysica, and vindicated by Aristotle, are —
1. The Maxim of Contradiction: It is impossible for the same thing to be and not to be; It is impossible for the same to belong and not to belong to the same, at the same time and in the same sense. This is the statement of the Maxim as a formula of Ontology. Announced as a formula of Logic, it would stand thus: The same proposition cannot be both true and false at the same time; You cannot both believe and disbelieve the same proposition at the same time; You cannot believe, at the same time, propositions contrary or contradictory. These last-mentioned formulae are the logical ways of stating the axiom. They present it in reference to the believing or disbelieving (affirming or denying) subject, distinctly brought to view along with the matter believed; not exclusively in reference to the matter believed, to the omission of the believer.
2. The Maxim of Excluded Middle: A given attribute either does belong, or does not belong to a subject (i.e., provided that it has any relation to the subject at all) — there is no medium, no real condition intermediate between the two. This is the ontological formula; and it will stand thus, when translated into Logic: Between a proposition and its contradictory opposite there is no tenable halting ground; If you disbelieve the one, you must pass at once to the belief of the other — you cannot at the same time disbelieve the other.
These two maxims thus teach — the first, that we cannot at the same time believe both a proposition and its contradictory opposite; the second, that we cannot at the same time disbelieve them both.[18]
[18] We have here discussed these two maxims chiefly in reference to Aristotle’s manner of presenting them, and to the conceptions of his predecessors and contemporaries. An excellent view of the Maxims themselves, in their true meaning and value, will be found in Mr. John Stuart Mill’s Examination of the Philosophy of Sir W. Hamilton, ch. xxi. pp. 406-421.
Now, Herakleitus, in his theory (a theory propounded much before the time of Protagoras and the persons called Sophists), denied all permanence or durability in nature, and recognized nothing except perpetual movement and change. He denied both durable substances and durable attributes; he considered nothing to be lasting except the universal law or principle of change — the ever-renewed junction or co-existence of contraries and the perpetual transition of one contrary into the other. This view of the facts of nature was adopted by several other physical philosophers besides.[19] Indeed it lay at the bottom of Plato’s new coinage — Rational Types or Forms, at once universal and real. The Maxim of Contradiction is intended by Aristotle to controvert Herakleitus, and to uphold durable substances with definite attributes.
[19] See ‘Plato and other Comp. of Sokr.’ I. i. [pp. 28-38].
Again, the theory of Anaxagoras denied all simple bodies (excepting Noûs) and all definite attributes. He held that everything was mingled with everything else, though there might be some one or other predominant constituent. In all the changes visible throughout nature, there was no generation of anything new, but only the coming into prominence of some constituent that had before been comparatively latent. According to this theory, you could neither wholly affirm, nor wholly deny, any attribute of its subject. Both affirmation and denial were untrue: the real relation between the two was something half-way between affirmation and denial. The Maxim of Excluded Middle is maintained by Aristotle as a doctrine in opposition to this theory of Anaxagoras.[20]
[20] Ibid. [pp. 49-57].
Both the two above-mentioned theories are objective. A third, that of Protagoras — “Homo Mensura� — brings forward prominently the subjective, and is quite distinct from either. Aristotle does indeed treat the Protagorean theory as substantially identical with that of Herakleitus, and as standing or falling therewith. This seems a mistake: the theory of Protagoras is as much opposed to Herakleitus as to Aristotle.