Excluded Middle, Principle of the

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The principle of the excluded middle is stated by aristotle: "There cannot be an intermediate between contradictions, but of one subject we must either affirm or deny any one predicate" (Meta. 1011b 2324). His treatment of this proposition is in Book Γ of the Metaphysics, which is devoted largely to the manifestation and defense of the first principles of demonstration.

Aristotle's Explanation. The proposition is made clear from the definitions of the true and the false, for it is false to say of what is that it is not, or of what is not that it is; and it is true to say of what is that it is, and of what is not that it is not. If anyone says something is, he either says something true or something false. If he is saying something true, the thing is; if he is saying something false, the thing is not. The same applies if he says something is not. Either the affirmation or the negation is true. The man who holds to an intermediate between contradictions does not grant that one must say of a being that it is or is not, nor of a nonbeing that it is or is not.

Contradictory Opposition. The basis for the principle of the excluded middle is found in the notion of contradictory opposites (see opposition). Things that are opposed as affirmation and negation are such that it is always necessary that one should be true but the other false (Cat. 13b 13). Since contradiction is a relation between terms opposed as affirmation and negation, it is an opposition between being and nonbeing; thus it makes no difference whether the subject actually exists or not. For example, it always is true or false that Socrates is ill. If Socrates actually exists, he is either ill or not. If he does not actually exist, it is false to say he is ill and true to say he is not ill, for he cannot be ill if he does not exist. Contradictory opposites are therefore quite different from contrary opposites that demand a common subject. The contradictory of "Socrates is ill" is not "Socrates is well," but "Socrates is not ill." Contradictory opposition is between being and nonbeing expressed in affirmative and negative statements; it is between being and nonbeing absolutely, and not within a genus. Either of the two opposites may be true or false, but not both true or both false at the same time.

Future Contingents. When two enunciations are in contradictory opposition, is it necessary that one be true and the other false? This question has concerned logicians and philosophers since the time of Aristotle (Interp. 18a 2819b 4). His answer is that propositions about the past or the present must be true or false; likewise, for any universal proposition and its contradictory one must be true and the other false; but for a singular proposition about the future, the case is different. For propositions about the past or the present there is a state of affairs against which the truth or falsity of a proposition can be measured, and this is true regardless of whether the propositions are about necessary or contingent matter. But for singular propositions about the future, there is no state of affairs that can be enunciated truly or falsely. Although singular propositions in necessary or impossible matter do have a determinate truth or falsity, future singular propositions in contingent matter do not.

To illustrate his discussion, Aristotle used the now celebrated example of the sea battle that will or will not take place tomorrow (ibid. 18b 24). If it is true now that the sea battle will take place tomorrow and false that it will not take place, a deterministic position is assumed that eliminates contingency and makes all events necessary. Aristotle rejects such a position, "for there is a difference between saying that that which is, when it is, must needs be, and simply saying that all that is must needs be, and similarly in the case of that which is not" (19a 2427). Rather, he says that neither contradictory is determinately true or false now. This allows for no intermediate between "the sea battle will take place" and "the sea battle will not take place." One or the other of these contradictions will be true and the other false, but neither is so now. The reason lies in the fact that at present the sea battle, however likely it may be, exists only potentially, and there is a possibility that it will never actually come to be.

Other Interpretations. Because of their strict determinism, Stoics held the determinate truth or falsity of every proposition, eliminating the possibility of future contingents. epicurus, on the other hand, is reported by Cicero (De fato 21) to have denied that every proposition is true or false.

St. thomas aquinas, in a commentary that exceeds the limits of mere exposition, sheds considerable light on Aristotle's position by analyzing the reasons that account for contingency, namely, the potentiality inherent in matter and the freedom of the human will (In 1 perih. 13, 14,15).

Some recent historians of logic have alleged that Aristotle called the principle of the excluded middle into question in that "he will not allow it to be valid for future contingent events" (Bocheński, 63) or that he tried to hold the principle of the excluded middle while denying the principle of bivalence (W. C. and M. Kneale, 4748). The Kneales call the principle that every statement is true or false the principle of bivalence, and formulate the principle of excluded middle, "'Either P or not-P,' where 'P' marks a gap into which a declarative sentence may be inserted." However, they regard the two principles as equivalent and consider Aristotle's treatment of singular future contingents a mistake. The Kneales' delineation of the mistake is too involved for condensation here; while they regard it as of considerable philosophical interest, they deem it of no logical importance.

Scholz (8688), following Moritz Schlick, takes the position that "in every proposition there inheres truth or falsity as a timeless property." A statement such as "Event E will occur on such and such a day" is a timeless statement and is true or false now. But since the truth or falsity of that proposition cannot be calculated on the basis of propositions about present events, we cannot know whether the proposition is true until the point of time has passed. In his view the proposition is true whether we know it or not.

John Stuart Mill (183) denied that an assertion must be either true or false on the ground that there is a third possibility, the unmeaning. He said, for example, "Abracadabra is a second intention" is neither true nor false. F. H. bradley (155) countered by pointing out that a proposition without meaning is no proposition, and that if it does mean anything, it is either true or false.

See Also: first principles; contradiction, principle of; truth; falsity.

Bibliography: g. giannini, Enciclopedia filosofica, 4 v. (Venice-Rome 1957) 4:117172. j. m. baldwin, ed., Dictionary of Philosophy and Psychology, 3 v. in 4 (New York 190105; repr. Gloucester 194957). a. e. babin, The Theory of Opposition in Aristotle (Notre Dame, Ind. 1940). h. w. b. joseph, An Introduction to Logic (2d rev. ed. Oxford 1916). l. s. stebbing, A Modern Introduction to Logic (London 1930). w. c. and m. kneale, The Development of Logic (Oxford 1962). h. scholz, Concise History of Logic, tr. k. f. leidecker (New York 1961). i. m. bocheŃski, A History of Formal Logic, tr. i. thomas (Notre Dame, Ind. 1961). f. h. bradley, The Principles of Logic, 2 v. (2d rev. ed. London 1922) v.1. j. s. mill, A System of Logic (New York 1904). n. rescher, Studies in the History of Arabic Logic (Pittsburgh 1964) ch. 5.

[h. j. dulac]