lies in this, that it limits the sphere of the thinkable in relation to affirmation. It determines that of the two forms. given by the laws of Identity and Contradiction, and by these laws affirmed as those exclusively possible, the one or other must be affirmed as necessary.' Hamilton seems to have fallen into the error of supposing that the law of Excluded Middle is a principle of Apodicticity, and gives necessary results. It necessitates the affirmation of one or other of the opposed contradictories. It does not affirm the one or other to be necessary. Besides, the formula which Hamilton uses is really the formula for the joint axiom of Contradiction and Excluded Middle, and does not express the latter purely. Cf. § 79.

2

J. S. Mill thinks that this law is one of the principles of all reasonings, being the generalisation of a process which is liable to be required in all of them. It empowers us to substitute for the denial of two contradictory popositions the assertion of the other two. He denies in his Logic the necessity and universality of the law, and says that it is not even true without a large exception. A predicate must be either true or false, provided that the predicate be one which can in any intelligible sense be attributed to the subject. Between the true and the false there is always a third possibility-the unmeaning. There are many valuable remarks in the pages Mr. Mill has given to the discussion of these laws, and had he not been hampered by his empirical theory of the origin of all knowledge, and his consequent theory of the supposititious nature of demonstrative science, he would have approached very nearly to the doctrine laid down in the text. Had he only pursued the theory laid down in discussing propositions-that they express real relations, he would have arrived at it. But there always seems to be a double view of Logic before Mr. Mill, and he shifts from the one to the other. On the one view Logic is a theory of knowledge, on the other it is almost a theory of naming. Examination of Sir W. Hamilton's Philos. 3rd ed. p. 473. 2 i. 309.

The two views come out most clearly in the chapters on propositions. Propositions in general describe facts, but Definitions describe names. In what is said of the laws of thought in his Logic the former view predominates; in what is said in the Examination, &c., the latter.

A. Bain confuses the opposition of predicates as contradictories, with the so-called contradictory opposition of judgments, to the extent that he makes the one grow out of the other; while they are in no way related. He thus makes the law of Excluded Middle an 'incident of partial or incomplete contrariety;' and says: It is too much honoured by the dignity of a primary law of thought."]

79. The axiom of Contradiction and the axiom of Excluded Middle may be comprehended in the formula: A is either B or is not в. Any predicate in question either belongs or does not belong to any subject; or-of judgments opposed as contradictories to each other, the one is true and the other false; or-To every completely distinct question understood always in the same sense, which has to do with the possession of a definite attribute by a definite subject, yes or no must be answered. These formulae contain the axiom of Contradiction, for they posit two contradictory members, and assert that the affirmation and denial of the same cannot both be true; A is either B, or is not B. They also contain the axiom of Excluded Third, for they posit only two mutually exclusive members, and assert, that any third besides affirmation and negation. is inadmissible, and that both are not false, but one of the two is true,—A is either B or is not B; there is no third. The comprehension of the axioms of Contradic1 Deductive Logic, p. 17.]

tion and Excluded Third in the foregoing formula may be called the PRINCIPLE OF CONTRADICTORY DISJUNCTION (principium disiunctionis contradictoriae).

A suitable statement of the question is again the natural presupposition of the application of this principle.

The transference of the denial to the predicate 'A is either B or non-B,'-is not false, provided that, by non-в, only contradictory opposition be understood. It is a useless artifice, however, and easily gives rise to a false meaning in the contrary opposite.

The simplest metaphysical formula of the principle of Contradictory Disjunction is found as early as in Parmenides,' ěσtiv ἢ οὐκ ἔστιν. It is here used only in the sense of the axiom of Contradiction to reject the common truth of the assertion of Being and Not-Being. Being and Not-Being cannot exist together, the one excludes the other.

5

Aristotle, on the other hand, uses the comprehensive formula mostly in the sense of the axiom of Excluded Third.2 ἀλλὰ μὴν οὐδὲ μεταξὺ ἀντιφάσεως ἐνδέχεται εἶναι οὐθέν, ἀλλ ̓ ἀνάγκη ἢ φάναι ἢ ἀποφάναι ἓν καθ' ἑνὸς ὁτιοῦν.3 πᾶν ἢ φάναι ἢ ἀποφάναι ἀναγκαῖον. ἐπὶ τῆς καταφάσεως καὶ τῆς ἀποφάσεως ἀεὶτὸ ἕτερον ἔσται ψεῦδος καὶ τὸ ἕτερον ἀληθές. τὸ δ ̓ ἅπαν φάναι ἢ ἀποφάναι ἡ εἰς τὸ ἀδύνατον ἀποδειξις λαμβάνει. Aristotle tried to deduce the axiom from the definitions of truth and untruth, on the ground of the presupposed impossibility that the same could be and not be. Every judgment (because it is a subjective assertion about objective existence) must fall under one of four forms of combination: denying what exists, affirming what does not exist; affirming what exists, denying what does not exist. The first two of these are false, the last two are true (for in the former the thought does not correspond with the actual fact; in the latter it does). The one assertion is true and the other false on presupposition of Being, and also so on presupposition of Not-Being. And in 1 Fragm. vs. 72, ed. Mullach; ap. Simplic. ad Arist. Phys. fol. 31 B. 2 Metaph. iv. 7, § 1. 4 Categ. c. x. 13 в, 27.

Ib. 8, § 6.

5 Anal. Post. i. 11.

every case either the affirmation or the negation is true, and, therefore, since truth is what we aim at, ή φάναι ἢ ἀποφάναι ȧvaykalov. Both cannot be false and a Third or Middle true. No place is left for a Middle. For a Middle, if it be true, or even thinkable, and have the reference to truth and falsehood, which belongs essentially to every judgment, must itself be one of that combination of members, which it cannot be according to its very notion. For in the Middle neither what exists nor what does not exist is affirmed or denied. It is in this way that the incompletely expressed reasoning of Aristotle against the Middle or Third must be completed.'

Leibniz places the negative form- A proposition is either true or false,' by the side of the affirmative form of the primitive, identical, rational truth-'Everything is what it is.' He calls this axiom the principle of Contradiction, and divides it into the two axioms which he includes in it- That a proposition cannot be true and false at the same time;' andThat there is no mean between the true and the false,' or rather-It is not possible that a proposition can be neither true nor false. In the same way Leibniz3 calls, Principe de la Contradiction,' the one 'which asserts that of two contradictory propositions the one is true and the other false.' Leibniz, therefore, understands by the Principle of Contradiction that axiom which includes both what is usually called the Axiom of Contradiction and the Axiom of Excluded Third.

Wolff' enunciates the formulae 'quodlibet vel est, vel non est;' 'propositionum contradictoriarum altera necessario vera, altera necessario falsa,' and says, 'patet per se, eidem subiecto A idem praedicatum B vel convenire, vel non convenire.' Many, both of the earlier and of the later logicians, have wrongly believed the formula-A is either B or not B, which includes both the axiom of Contradiction and of Excluded Third, to be the proper and simple expression of the axiom of Excluded Third."

§ 80. The foregoing Axioms are not to be applied to judgments whose PREDICATES stand to each other in the relation of CONTRARY opposites (like positive and negative quantities). In this relation it is possible under certain presuppositions, that (a) both judgments be false; and that (b) both judgments be true.

Both may be false:

1. When that notion which is superordinate to the two predicates opposed to each other as contraries as their common genus-notion does not belong to the subject as its predicate. (Kant called this relation Dialectical Opposition.)

2. When that genus-notion belongs to the subject, but comprehends under it, besides the two predicates opposed as contraries to each other, other species-notions. In this last case the axiom of the Third lying as a mean between two contrary opposites finds application (principium tertii intervenientis inter duo contraria).

Both may be true:

When the subject denotes an object, which is neither absolutely simple nor yet a mere aggregate, but is a synthetic unity of manifold determinations. When some of these determinations or attributes stand in the relation of contrary opposites to each other, the axiom of Coincidence may be applied to them (principium coincidentiae oppositorum). All development by strife and union of opposites rests on this principle.

Judgments, whose predicates are opposed to each other as contraries (§ 53)-e.g. Caius is happy, Caius is sad-are to be strictly distinguished from judgments which are contrarily opposed to each other as judgments (§ 72)—e.g. all men are