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the inclusion of one class in the other, but the identity of group with group. The proposition becomes an equation of subject and predicate, and the significance of this change will be fully apparent only to those who see that logical science thus acquires a point of contact with mathematical science. Nor is it only in a single point that the two great abstract sciences meet. Dr. Boole's remarkable investigations prove that, when once we view the proposition as an equation, all the deductions of the ancient doctrine of logic, and many more, may be arrived at by the processes of algebra. Logic is found to resemble a calculus in which there are only two numbers, o and I, and the analogy of the calculus of quality or fact and the calculus of quantity proves to be perfect. Here, in all probability, we shall meet a new instance of the truth observed by Baden Powell, that all the greatest advances in science have arisen from combining branches of science hitherto distinct, and in showing the unity of principles pervading them.1
9. And yet any one acquainted with the systems of the modern logicians must feel that something is still wanting. So much diversity and obscurity are no usual marks of truth, and it is almost 1 Baden Powell, "Unity of the Sciences," p. 41.
incredible that the true general system of inference should be beyond the comprehension of nearly every one, and therefore incapable of affecting ordinary thinkers. I am thus led to believe that the true clue to the analogy of mathematics and logic has not hitherto been seized, and I write this tract to submit to the reader's judgment whether or not I have been able to detect this clue.
10. During the last two or three years the thought has constantly forced itself upon my mind, that the modern logicians have altered the form of Aristotle's proposition without making any corresponding alteration in the dictum or self-evident principle which formed the fundamental postulate of his system. They have thus got the right form of the proposition, but not the right way of using it. Aristotle regarded the proposition as stating the inclusion of one term or class within another; and his axiom was perfectly adapted to this view.
The so-called Dictum de omni is, in Latin phrase, as follows
Quicquid de omni valet, valet etiam de quibusdam et singulis.
And the corresponding Dictum de nullo is similarly
Quicquid de nullo valet, nec de quibusdam nec de singulis valet.
In English these dicta are usually stated somewhat as follows:
Whatever is predicated affirmatively or negatively of a whole class may be predicated of anything contained in that class. Or, as Sir W. Hamilton more briefly expresses them, What pertains to the higher class pertains also to the lower.1
These dicta, then, enable us to pass from the predicate to the subject, and to affirm of the subject whatever we know or can affirm of the predicate. But we are not authorized to pass in the other direction, from the subject to the predicate, because the proposition states the inclusion of the subject in the predicate, and not of the predicate in the subject. The proposition,
All metals are elements,
taken in connexion with the dictum de omni authorizes us to apply to all metals whatever knowledge we may have of the nature of elements, because metals are but a subordinate class included among the elements; and, therefore, possessing all the properties of elements. But we commit an obvious fallacy if we argue in the opposite direction, and infer of elements what we know only of 1 "Lectures on Logic," vol. i. p. 303.
metals. This is neither authorized by Aristotle's dictum, nor would it be in accordance with fact. Aristotle's postulate is thus perfectly adapted to his view of the nature of a proposition, and his system of the syllogism was admirably worked out in accordance with the same idea.
II. But recent reformers of logic have profoundly altered our view of the proposition. They teach us to regard it as an equation of two terms, formerly called the subject and predicate, but which, in becoming equal to each other, cease to be distinguishable as such, and become convertible. Should not logicians have altered, at the same time and in a corresponding manner, the postulate according to which the proposition is to be employed? Ought we not now to say that whatever is known of either term of the proposition is known and may be asserted of the other? Does not the dictum, in short, apply in both directions, now that the two terms are indifferently subject and predicate?
12. To illustrate this we may first quantify the predicate of our own former example, getting the proposition,
All metals are some elements,
where the copula are means no longer are contained among, but are identical with; or availing
ourselves of the sign in a meaning closely analogous to that which it bears in mathematics, we may express the proposition more clearly as,
All metals = some elements.
It is now evident that whatever we know of a certain indefinite part of the elements we know of all metals, and whatever we know of all metals we know of a certain indefinite part of the elements. We seem to have gained no advantage by the change; and if we are asked to define more exactly what part of the elements we are speaking of, we can only answer, Those which are metals. The formula
is a more clear statement of the same proposition with the predicate quantified; for while it asserts an identity it implies the inclusion of metals among elements. But it is an accidental peculiarity of this form that the dictum only applies usefully in one direction, since if we already know what metals are we must know them to be metallic elements, the adjective metallic including in its meaning all that can be known of metals; and from knowing that metals are metallic elements we gain no clue as to what part of the properties
all metallic elements