ject; No subject can have a predicate that contradicts it;" What is contradictory is unthinkable; " No object can be thought under contradictory attributes; " The same thing cannot be A and non-A: this room cannot be both hot and cold." As this law enjoins the absence of contradiction as the indispensable condition of thought, it ought rather to be called the Law of Non-contradiction." The formula is: A is not A=0; or, A is not non-A. Examples which, if taken literally, violate the law are: "Dotage is infancy in old age;" "His left hand is most dexterous;" "The blind see, the deaf hear, the dumb speak," etc.; "However unwilling the choice, he was compelled to volunteer;" "Since the war, all values have risen;" "Two kinds of individuals prepare extempore speeches, fops and fools;" "Nothing in this life is true;" "The decomposition of the elements;" "We want nothing but silence, and but little of that." Each of the foregoing examples is a logical paradox, a self-contradiction; each violates the law, and is a felo de se. By a fundamental law of mind, which Bain calls the Law of Relativity, every notion has an opposite or counter notion, and only by virtue of the one can the other be conceived. To the straight line there is opposed the not-straight line, or crooked line; to good is opposed evil, and a knowledge of good is impossible to a mind not knowing evil. Hence the old scholastic maxim: Contrariorum eadem est scientia. Now these opposites cannot consist, their union is contradiction, and thorough-going consistency, as formulated in the Law of Contradiction, forbids it. Thus, when we affirm that this is a straight line, we must not also say that it is a crooked line; when we think an act good, we may not also think it evil. Our assertions, our thoughts, to be consistent, must not contradict each other. If they do, the thought is null, it destroys itself. Having made an assertion, we are to abide by that. Affirmations not self-consistent are unintelligible. But the principle of contradiction carries us one step further. An affirmation being made, it not merely forbids us to affirm also its contradictory, but it authorizes us, or requires us, to pronounce the contradictory false; i. c., to deny, of an object of thought, its contradic Aristotle, who says this is by nature the principle of all other axioms.—Metaph. IV (r), lii. 10 Kant's Critique of Pure Reason. See Meiklejohn's transl. p. 115. "Hamilton's Logic, p. 58. 13 Bain's Logic, p. 16. 12 Mansel's Prolegomena Logica, p. 167. tory. Accordingly, the principle may be enunciated thus: Of two contradictories, one must be false. E. g., "This straight line is not crooked;" "This good act is not evil;" "No chastisement is joyous;" "Francis Bacon was not Roger Bacon;" "A dishonest man is not trustworthy." If all diamonds are precious, then to say that some or any diamonds are not precious is false. Whatever is repugnant, opposite, contradictory, to a notion must be denied of it. The Laws of Identity and Contradiction are co-ordinate. Neither can be deduced as a second from the other as first. In every such attempt the evolved secondary is unavoidably presupposed, which is petitio principii." The two have, however, been identified by many eminent philosophers, as Leibnitz, Wolf, Kant, Herbart. And Hamilton says, "The two laws are essentially one, differing only by a positive and negative form." Perhaps the two may be fairly summed in the statement: All thought must be self-consistent. § 4. The third is the Law of Excluded Middle. Its logical significance is that it limits the thinkable in relation to affirmation; for it determines that of the two forms given in the first two laws, the one or the other must be affirmed as necessary. No middle ground, no third affirmation, being possible, one or the other must be true. Hence the names: Lex exclusi medii aut tertii inter duo contradictoria; Principium contradictionis affirmativum. We enunciate it thus: Of two contradictories, one must be true. Either a given judgment must be true, or its contradictory: there is no middle course.' Of two contradictories, one must exist in every subject." The formula is: X is either A or non-A; one being sublated, the other must be posited. 17 A few examples will suffice: "Either it is true that God exists, or it is true that he does not exist;" "Man must be a free agent, unless his acts are necessitated;" "To be or not to be, that is the question;" "Infinite mercy offers salvation to all." In this last example the opposition is between bounds and no-bounds; bounds is denied in "infinite," and hence no-bounds must be affirmed, which is done in "offers salvation to all." The argument called Reductio ad absurdum is an application of this law. Of two alternatives it shows one to be absurd, hence the other must be true; for one proposition being false, 15 Shown in Hoffbauer's Logik, § 23. 17 Thomson's Outline, § 114. 10 Logic, p. 59. 18 Mansel's App. to Aldrich, p. 241. we are authorized or required by this law to pronounce its contradictory true. The Laws of Contradiction and Excluded Middle may be conveniently united in one statement, to which might be given the name "Law of Duality." It is the principle of strict logical division and disjunction." We may enunciate it thus: Of two contradictories, one must be true, the other false; Every predicate may be either affirmed or denied of every subject; Every assertion must be either true or false." This compound form is often mistaken by logical writers for the Law of Excluded Middle. So Goclenius: Oportet de omni re affirmare aut negare." Hamilton also. He gives for the Law of Excluded Middle: Of contradictory attributions, we can af firm only one of a thing; and if one be explicitly affirmed, the other is implicitly denied." This is the compound; the latter member is the principle of contradiction. His subsequent exposition, however, is correct. Bain clearly makes the mistake." So also Herbert SpenHe says the principle of Excluded Middle is: The appearance of any positive mode of consciousness cannot occur without excluding a correlative negative mode; and the negative mode cannot occur without excluding the correlative positive mode." cer. § 5. The Laws of Identity, Contradiction, and Excluded Middle are mutually complementary. "The object which I conceive is by the Law of Identity discerned as being that which it is, and by the Law of Contradiction is distinguished from that which it is not. these two correlatives must also be regarded as constituting between them the universe of all that is conceivable; for the distinction above made is not between two definite objects of thought, but between the object of which I think and all those of which I do not think. Non-A implies the exclusion of A only, and of nothing else, and thus denotes 10 The 'Αξίωμα διαιρετικών of the Greeks. 20 Mill questions the absolute truth of this axiom.-Logic, p. 205. He says that between the true and the false there is a third possibility, the unmeaning: e. g., "Acracadabra is a second intention," is neither true nor false. But is an unmeaning proposition any assertion at all? Its content is a vacuum. If unmeaning, it means nothing, says nothing. The third possibility, then, is nothing; or, there is nothing between the true and the false. See also Examination of Hamilton, ch. xxi. 21 Lex. Philosoph. p. 136. 23 Logic, p. 17. 22 Logic, p. 59. 24 25 the universe of all conceivable objects with that one exception." " In other words, A and non-A divide the universe between them, admitting no intermediate or third possibility, which is declared by the Law of Excluded Middle. By the Law of Identity, whatever is one is that one. By the Law of Contradiction, whatever is one is not the other. By Excluded Middle, every possible thing is either A or non-A. By the former, two contradictories cannot both be true; i. c., one must be false. By the latter, two contradictories cannot both be false; i. e., one must be true. to one. Many fruitless attempts have been made to reduce the three laws So intimate is their relation that each supposes the other; but, like the sides of a triangle, they are not the same, not reducible to unity, each having equal right to be considered first, and each, if considered first, giving, in its own existence, the existence of the other two. Accordingly every attempt to deduce either one from the others has failed. They are complementary, co-ordinate, distinct, and inseparable.' 26 § 6. Whatever violates either of these laws we feel to be impossible, not only in thought, but in existence. We cannot believe that anything can differ from itself, that anything can at once be and not be, that anything can neither be nor not be. We cannot but regard that as false and unreal which these laws condemn. They thus determine to us the sphere of impossibility, and that not merely in thought, but in reality; not only logically, but metaphysically. What is contradictory is inconceivable in thought and impossible in fact. But, on the other hand, it does not hold that what is thought in conformity with these laws is therefore true in reality; that whatever is conceivable in thought is actual, or even possible, in fact. For the sphere of thought is far wider than the sphere of reality, and no inference is valid from the correctest thinking of an object to its actual existence. What is conceivable conforms to the laws of thought, and Mansel's Prolegomena Logica, p. 168. 26 Hamilton's Logic, pp. 70 and 506. is said to be logically possible, i. e., possible in thought; and this is true of many things that are impossible in fact. Pure mathematics deals exclusively with mere logical possibilities. That the stars may fall on the earth is physically impossible; that revenge may be a duty is ethically impossible. But both are conceivable; they may be represented in thought; they are logically possible. I may think Waterloo a fiction, or Christianity a failure, but this conceivability is no evidence that they are so. While, then, these laws are the highest criterion of the non-reality of an object, they are no criterion at all of its reality; and they thus stand to existence in a negative, and not in a positive, relation. Says Kant, "The principle of contradiction is a universal but purely negative criterion of all truth." " And this holds equally of all the proximate and special applications of these laws; that is, of the whole of Logic. Our science, then, in its relation to other sciences, is not a positive criterion of truth; it can only be a negative criterion, being conversant with thoughts and not with things, with the possibility and not the reality of existence. We have referred to the psychological Law of Relativity. Some eminent German philosophers have held that the human mind is competent to the cognition of the absolute, or that which has no relation, and have elaborated thereon extensive systems of philosophy. This Philosophy of the Absolute can proceed only upon a more or less complete denial of the primary laws of thought. Fichte and Schelling admit the Law of Identity, but deny the two others, "the empirical antagonism between the Ego and the Non-ego being merged in the identity of the absolute Ego." Hegel regards all the laws as valid, but only for the finite Understanding, they being inapplicable to the higher processes of the Reason. The eclecticism of Cousin attempts the cognition of the absolute without repudiating the laws of Logic. It is therefore at once involved in undeniable contradictions from which there is no escape. § 7. The principle of Sufficient Reason, or Determinant Reason, has been laid down as a fourth primary law of thought. It is enunciated thus: Every judgment must have a sufficient ground for its assertion. It was first distinctly enounced by Leibnitz, who made it, together with the principles of Identity and Contradiction, the basis of his Logic. Kant adopted it, regarding Contradiction as the crite |