View from the Dunes with Beach and Piers, Domburg, Piet Mondrian, 1909
Aristotle devotes the entirety of the fourth book of his Metaphysics to developing and defending what philosophers and theologians have traditionally considered to be the primary and self-evident principle of being and thought. He ends up stating the principle in two ways: (1) “The same attribute cannot at the same time belong and not belong to the same subject in the same respect” and (2) “No one can suppose the same thing to be and not to be.” In traditional logic and philosophy, this principle was called the “principle of non-contradiction,” while in modern times it has been called the “law of contradiction.” In what follows, I will refer to it for the sake of convenience simply as “the Principle.”
In applying the Principle primarily to the natures of things themselves, Aristotle was correcting a long tradition of philosophers in the world of the ancient Greeks. “There are some who... assert that it is possible for the same thing to be and not to be, and say that people can judge this to be the case. And... many writers about nature use this language.”1 As W. D. Ross summarizes the matter, “They maintain the possibility of contradiction 1) in fact—‘A may be both B and not B,’ and 2) in belief—‘a man may judge that A is B and also that A is not B.’”2 In the Metaphysics, Aristotle chooses to defend the Principle “in belief”—that is, by showing how those who argue against it end up contradicting themselves. Accordingly, his argument took the form of a reductio ad absurdum.
“Aristotle has in his defense of the law of contradiction… used an argument derived from the necessity of a fixed meaning for every term.”3 If the opponent is to argue at all, he must assume that his words keep the same meaning throughout—that is, he must “say something which is significant both for himself and for another.”4 But if he does this, he concedes the truth of the Principle even while attempting to disprove it. “Again, if contradictory statements are all true of the same subject at the same time, evidently all things will be one. For the same thing will be a trireme, a wall, and a man, if of everything it is possible either to affirm or to deny anything.”5 “If the meanings are unlimited, no account can be given of the thing. Not to mean one thing is to mean nothing, and if words mean nothing rational intercourse with others is destroyed, and even with oneself, for if we do not think one thing we do not think at all.”6 Aristotle goes so far as to state several times that a person acting in such a way is “no better than a vegetable.”7
The law of identity, implicit in ancient and medieval philosophy, did not assert the mathematical principle “A is A” but rather the logical principle that “A is B,” such that B states the essence of A. In other words, B states explicitly what A is. Thus, “Every human being is an animal” does not say that “human being” is “animal” but that the definition of “animal” is contained in the definition of “human being.”
While contradiction by negating absolutely the nature of a thing is the strongest of the species of opposition, the other main species, that of contrary opposition, presupposes a reconciling third term that allows the contrary terms to participate in a common nature. Contrary opposites are both like and unlike: spiders are “like” flies insofar as both are animals (i.e., both share the same remote genus). And they are unlike not only in appearance but because each has a specific difference that sets it apart in its own species within that genus. Virtue does not contradict vice, for there are degrees in each that approximate the other; one need not be vicious if one is lacking in virtue, even severely so. “Wherever there is a metaxy,8 we can observe it changing into the extremes between which it lies (for change is just from extreme to extreme, from extreme to intermediate, or from intermediate to extreme).”9 But if we say that “a fly is a spider” or “an elephant is a palm tree,” we unqualifiedly deny the nature of the terms. “Therefore there is no such thing as a metaxy of contradictories in the sense of a mere neutral between them.”10 There is nothing in-between two contradictories, and thus we have the principle of the excluded middle, which for Aristotle is a way of expressing the absolute difference between contradictory and contrary opposition: “There cannot be an intermediate between contradictories, but of one subject we must either affirm or deny any one predicate.”11
The English term principle, itself deriving from the Latin principium, translates the Greek archē, which means both “first” and “rule.” We have seen how it serves as a rule, and it is first not because it comes prior in time but because by it everything capable of being known can be given a logos, an account of its essential nature, of what it is. But this means that the Principle, as constituting the ultimate criterion for certainty, is even prior to our knowledge of truth. The Principle is the strictest and most universal limit to thinking and being, applicable everywhere and at all times, present seminally or virtually in the hidden part of the mind—Aristotle went so far as to maintain there was a separate faculty in the mind, “nous,” that intuited the Principle—and unfolding into actuality with every mental or spoken judgment, no matter how seemingly trivial or important. Those who deny it “seek a reason for things for which no reason can be given; for the starting-point [the archē or Principle] of demonstration is not demonstration.”12 Ultimately, then, to every existing thing there must be an abiding essential nature, and to each word there must be an abiding essential definition. To deny the former is to violate the Principle in fact, to deny the latter is to deny the Principle in belief.
For us who live today, who have become accustomed by the authority of science to believe many fantastic things, Aristotle’s Principle may seem trite. Indeed, modern philosophers have denied its primacy and/or denied its legitimacy. What great scientific advance does it provide or even promise? What effort of the mind does it cost to understand it? That everyone can understand it immediately upon trying to violate it seems rather to be the best proof of its unimportance. However, since the infancy of science, one can witness how great a challenge it was for philosophers to arrive at a convincing account of how there could be certain knowledge of the world. When one seeks for absolute certainty, the stability one was accustomed to seeing in nature seems to evaporate into a constant state of flux. Aristotle expresses the desire of the early scientists by the analogy of a soldier who makes a stand amid a route on the battlefield, making it possible for others to take a stand and eventually regain the ground they lost: in the battle to achieve stable knowledge amid the flux of mutable phenomena, the philosopher achieves the first “universal”—that is, the first stage of conceptual knowledge that allows others to do the same and eventually achieve science.