Now Logic does not take account of this implication, and nothing has contributed more to bring upon it the reproach of quibbling. In Logic, to deny a quality is simply to declare a repugnance between it and the subject; negation is mere sublation, taking away, and implies nothing more. Not-b is entirely indefinite: it may cover anything except b.
Is Logic then really useless, or even misleading, inasmuch as it ignores the definite implication of negatives in ordinary thought and speech? In ignoring this implication, does Logic oppose this implication as erroneous? Does Logic shelter the quibbler who trades upon it? By no means: to jump to this conclusion were a misunderstanding. The fact only is that nothing beyond the logical Law of Contradiction needs to be assumed for any of the processes of Formal Logic. Aristotle required to assume nothing more for his syllogistic formulæ, and Logic has not yet included in its scope any process that requires any further assumption. "If not-b represent everything except b, everything outside b, then that A is b, and that A is not-b, cannot both be true, and one or other of them must be true."
Whether the scope of Logic ought to be extended is another question. It seems to me that the scope of Logic may legitimately be extended so as to take account both of the positive implication of negatives and the negative implication of positives. I therefore deal with this subject in a separate chapter following on the ordinary doctrines of Immediate Inference, where I try to explain the simple Law of Thought involved. When I say that the extension is legitimate, I mean that it may be made without departing from the traditional view of Logic as a practical science, conversant with the nature of thought and its expression only in so far as it can provide practical guidance against erroneous interpretations and inferences. The extension that I propose is in effect an attempt to bring within the fold of Practical Logic some of the results of the dialectic of Hegel and his followers, such as Mr. Bradley and Mr. Bosanquet, Professor Caird and Professor Wallace.[10]
The logical processes formulated by Aristotle are merely stages in the movement of thought towards attaining definite conceptions of reality. To treat their conclusions as positions in which thought may dwell and rest, is an error, against which Logic itself as a practical science may fairly be called upon to guard. It may even be conceded that the Aristotelian processes are artificial stages, courses that thought does not take naturally, but into which it has to be forced for a purpose. To concede this is not to concede that the Aristotelian logic is useless, as long as we have reason on our side in holding that thought is benefited and strengthened against certain errors by passing through those artificial stages.
[Footnote 1:] The first statement of the Law of Identity in the form Ens est ens is ascribed by Hamilton (Lectures, iii. 91) to Antonius Andreas, a fourteenth century commentator on the Metaphysics. But Andreas is merely expounding what Aristotle sets forth in iii. 4, 1006 a, b. Ens est ens does not mean in Andreas what A is A means in Hamilton.
[Footnote 2:] τὸ γὰρ αὐτὸ ἅμα ὑπάρχειν τε καὶ μὴ ὑπάρχειν ἀδύνατον τῷ αὐτῷ καὶ κατὰ τὸ αὐτὸ, . . . αὕτη δὴ πασῶν ἐστὶ βεβαιοτάτη τῶν ἀρχῶν. iii. 3, 1005b, 19-23.
[Footnote 3:] Hamilton credits Andreas with maintaining, "against Aristotle," that "the principle of Identity, and not the principle of Contradiction, is the one absolutely first". Which comes first, is a scholastic question on which ingenuity may be exercised. But in fact Aristotle put the principle of Identity first in the above plain sense, and Andreas only expounded more formally what Aristotle had said.
[Footnote 4:] Μεταξὑ ὰντιφάσεως ἐνδέχεται εἶναι οὐθέν, ἀλλ᾿ ἀνάγκη ἢ φάναι ἢ ὰποφάναι ἒν καθ᾿ ἑνὸς ὁτιοῦν. Metaph. iii. 7, 1011b, 23-4.
[Footnote 5:] Prof. Caird's Hegel, p. 138.
[Footnote 6:] See Venn, Empirical Logic, 1-8.