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satisfies Jevons. This he has adopted, and, as one instantly sees, it may be carried on without saying anything about expansion, and without putting the development in the form of an equation. For convenience, we may always use the same letters, taking as many as we need in regular alphabetical order, and denoting positive terms by capitals, and their negatives by small letters. In our example, let A= animal, B= mian, C= rational, and, instead of developing only with reference to two terms, expand with reference to the three, and our constituents are eight in number, as follows: 1,
A Bc. 3,
a Bc. 7,
abc. Making our trial references to our premise, “ all men are all the rational animals," 2, 3, 5, 6 strike out, and, selecting the terms left containing A, we have for animals only animals, men rational, and animals, not-men, not-rational, the same as before. If using, with Boole, the principle of quantification of the predicate, we express our premises in these same letters, the making of trial-references becomes purely mechanical, and thus Boole's theorem gave rise to Jevons' interesting logical machine. This, as a result, by the way, is certainly very charming, but the end and aim, a genuine satisfactory algebra of logic, should be kept steadily in view. It is overlooking this that makes even such an acute critic as Mr. Venn blame Boole for giving to the last process we shall mention, the process of getting rid of any terms we choose from our equations, the name “ Elimination.” Says Mr. Venn: “In each case no doubt a term disappears from the result, but the meaning and consequences of its disappearance are altogether distinct.” Of course they are, but this is matter of interpretation, and to name the formal processes of a symbolic algebra according to interpretation would be in the highest degree unwise.
Here again our law of duality or simplicity, x-=X, comes to our aid and makes the problem of elimination resolvable under all circumstances alike. In common algebra there exists a definite connection between the number of independent equations given and the number of symbols of quantity which it is possible to eliminate from them ; but, in the algebra of logic, from even a single equation an-indefinite number of terms may be eliminated.
Here we will pause. We are now in position to see how it is that Boole’s Logical Method can give an absolutely general solution to the final problem of practical logic. Its mode of application to every possible case is evident from the analogy of common algebra, and we may refer to the book itself for examples, instead of taking any here of sufficient intricacy to give any adequate idea of its astonishing grasp and power.
We have made no attempt at a complete presentation of the system. Our desire has been to call attention to the principles which rendered it possible, to show where its imperfection lies, to throw light on those points where his readers have been most apt to go astray, and to heighten the interest beginning to be widely shown in a truly wonderful work.
NOTES AND DISCUSSIONS.
SONNET TO THE VENUS OF MILO.
O peerless marble! bold had been the thought,
When thou in nature's formless grasp didst lie,
Olympian glory, grace, and majesty.
Thee into being; one to whom the sky
Flood, forest, vale, declared harmoniously
For the great thoughts which pulsed within his soul.
And knew her glory was her lord's control.
B. E. S.
EMANUEL HVALGREN'S SYSTEM.
(We have received, from the philosopher above named, a syllabus of his lecture on the “Being and Existence of God and the World.” In Vol. VIII, p. 285, we have noticed his “Theocosmic System.”-Ed.]
A. THE IDEA SPIRIT: THE GODHEAD (ABSOLUTE FREEDOM).
Arguments for the Existence and Essence of God.
1. If God is not, He must have Freedom not to be. God is not; therefore He must have Freedom not to be.
2. If God is, He must have Freedom to be. God is; therefore He must have Freedom to be.
3. From this it follows that Freedom is the ground and condition for the non-being, as well as the existence, of God; and, consequently, higher than the common notion of God, whether as merely an unconscious abstractum or as self-consciousness (personality).
4. But, as not any notion can be higher than God, and Freedom is demonstrated to be the highest notion or principle, Freedom itself is God.
5. These arguments will, therefore, remain valid as long as the logical and mathematical laws of thought and nature are valid. And, if these should be suspended by a higher law, this, again, must have Freedom for its presupposition, and, consequently, be Freedom itself.
B. SPIRIT: THE WORLD (RELATIVE FREEDOM). Arguments for the Existence and Essence of the World. These resemble the foregoing, and, consequently, the World is in absolute Unity and Identity with Freedom.
EMANUEL HVALGREN. WARBERG, SWEDEN, August 15, 1877.
NOTES ON HEGEL AND HIS CRITICS.
We cannot help believing in the reality of pure thought, Hegel argues, in the Encyclopædia, no matter how thoroughly we may have schooled ourselves in the Cartesian scepticism. The will to think purely is all that is required of the beginner at the outset of the logic. Though it prove itself identical with being, pure thought is always the logical prius. Because it is first, and because, as any logical beginning must be, it is immediate, it is best represented as objective-as something given, to be observed or speculated, rather than controlled or comprehended. Here, as being and as essence, it is the most real of all realities; in short, it is substance itself, in its most self-subsistent nature.
In the logic of notion pure thought becomes its own equipollent subject, constituting the world in which consciousness lives and moves, and hence is the most ideal of all ideas—now not merely metaphysical, but transcendent. It is pure thought which is latent and determining abstract, in Hegel's sense, through all the stages of the Phenomenology, and which becomes articulate and explicit in the Logic. Thus, as the Neo-Platonists said of the relation between the Old and New Testaments, so we may say of the Phenomenology and the Logic: In the first the last lies concealed; in the last the first stands revealed.
There is no jenseits to the logician who has reached the perfect entelecheia of für sich. The picture is the curtain which seemed to hide it. Pure thought, then, which seemed so easy because it is so spontaneous and inevitable, proves in the end infinitely hard, because, as Michelet explains, not only are all the phenomenal stages of consciousness presupposed, but because the universal whole of thought is involved by the severest logical necessity in its simplest act. Pure thought, then, is not so much a dominant category in Hegel's system as the warp, which does not in itself contribute to form or color, although through it all the categories are woven with harmonious and determinate sequence into ideal patterns of things.
Does Hegel's system require us to conceive of thought as pure in an improbable sense? This has been a central question in all Hegelian discussions. It seems evident that “ a presuppositionless beginning does not require us to forego the use of concrete predicates," or “ metaphors of sense and understanding,” in characterizing it, nor forbid us to recognize any of the previous determinations of thought as we proceed. Indeed, it is perhaps more necessary for the dialectic than for the deductive method that it pause and verify at every step. Even Rosenkranz insists that the logic needs modifications because this was not sufficiently done by Hegel. Indeed, this is necessary not merely for the didactic success of any system, but it is perhaps the highest philosophic motive, for no speculation was ever truly satisfying to the philosophical impulse, or even very convincing as a mere act of first intellection, before it was brought into manifold and harmonious relations to common thought and things. But, on the other hand, if what claims to be a pure geometry of thought is found to be merely description of particular objects of thought—if idola fori, or the Zeitgeist, or empirical science are found to have furnished centers about which thought has accreted, instead of crystallizing into its own free forms, then it is impure, in a sense fatal to many cherished results of Hegelism.
Space, in Hegel's system, is derived only in the philosophy of nature as the first result of the creative resolve of the absolute idea in its pure freedom to become objective to itself. It is thus the otherbeing of spirit, the external as such, and in itself, without farther determination. While later, space and time, by their own imminent dialectic, become, as sublated, matter. Before this, quantity and measure, and even attraction, repulsion, and mechanism, are all characterized in the logic as non-spacial. It is evident, without discussion, that Hegel is no mathematician, and that this description of the origin of space is inadequate to the most important of all logical transi