To turn next to the Laws of Contradiction and Excluded Middle. These also may be subjected to Casuistry, making clearer what they assert by showing what they do not deny.
They do not deny that things change, and that successive states of the same thing may pass into one another by imperceptible degrees. A thing may be neither here nor there: it may be on the passage from here to there: and, while it is in motion, we may say, with equal truth, that it is neither here nor there, or that it is both here and there. Youth passes gradually into age, day into night: a given man or a given moment may be on the borderland between the two.
Logic does not deny the existence of indeterminate margins: it merely lays down that for purposes of clear communication and coherent reasoning the line must be drawn somewhere between b, and not-b.
A difference, however, must be recognised between logical negation and the negations of common thought and common speech. The latter are definite to a degree with which the mere Logic of Consistency does not concern itself. To realise this is to understand more clearly the limitations of Formal Logic.
In common speech, to deny a quality of anything is by implication to attribute to it some other quality of the same kind. Let any man tell me that "the streets of such and such a town are not paved with wood," I at once conclude that they are paved with some other material. It is the legitimate effect of his negative proposition to convey this impression to my mind. If, proceeding on this, I go on to ask: "Then they are paved with granite or asphalt, or this or that?" and he turns round and says: "I did not say they were paved at all," I should be justified in accusing him of a quibble. In ordinary speech, to deny one kind of pavement is to assert pavement of some kind. Similarly, to deny that So-and-so is not in the Twenty-first Regiment, is to imply that he is in another regiment, that he is in the army in some regiment. To retort upon this inference: "He is not in the army at all," is a quibble: as much so as it would be to retort: "There is no such person in existence".
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.
Footnote 7: E.g., Hamilton, lect. v.; Veitch's Institutes of Logic, chaps, xii., xiii.
Footnote 8: The confusion probably arises in this way. First, these "laws" are formulated as laws of thought that Logic assumes. Second, a notion arises that these laws are the only postulates of Logic: that all logical doctrines can be "evolved" from them. Third, when it is felt that more than the identical reference of words or the identity of a thing with itself must be assumed in Logic, the Law of Identity is extended to cover this further assumption.
Footnote 9: E.g., Bosanquet's Logic, ii. 207.
Footnote 10: Bradley, Principles of Logic; Bosanquet, Logic or The Morphology of Knowledge; Caird, Hegel (in Blackwood's Philosophical Classics); Wallace, The Logic of Hegel.