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or individuals to which their several meanings are together applicable, and to the law that the order in which the symbols succeed each other is indifferent.”
Again, to form the aggregate conception of a group of objects consisting of partial groups, we use the conjunctions “6 and," "or.” Convening that the classes so joined are quite distinct, so that no individual is added to himself, we see that these conjunctions hold precisely the same position formally as the sign + in the ordinary algebra of number, and, therefore, are represented by that sign. As the order of addition is indifferent, we have x+y=y+x; and, from IV, z (x+y)=2x+zy. Again, to separate a part from a whole, we express in common language by the sign “ except” (-), as, “ All men except Asiatics." This is our minus. As it is indifferent whether we express excepted cases first or last, we have x-y=-y+x, and, from IV, (xy) = x—zy.
So we may at once affirm for our logical algebra the validity of the three general axioms :
1. Equals added to equals give equals.
Though each of these may be demonstrated for the algebra of logic entirely independently of even the existence of any such thing as the algebra of number, yet we see it actually turns out that, so far, the two algebras are formally identical. This may lead the reader to wish that this formal identity had held throughout, so that he might have interpreted his quantitative mathematics directly as so much logic, just as the same process may, under one scheme of interpretation, represent the solution of a question on the properties of numbers, under another, that of a geometrical problem, and under a third, that of a problem in dynamics or optics. But let me repeat that, if no different operative law had manifested itself, the algebra of logic, like that of number, would have been stopped short at the equation of the fifth degree, and so its general problem could never have been solved.
Just as the algebra of quaternions differs in one fundamental law from the algebra of number, namely, in its multi
plication being non-commutative, so that ab does not equal ba, so our algebra of logic differs in a law equally fundamental, and from this difference comes the power that, in it, every equation can be solved and every solution interpreted.
The real nature and unavoidable character of this law in our new algebra depend upon the general postulates of thought which we have given ; but, unfortunately, Boole, groping in the darkness of a dawning subject, introduced the matter upside down, and so was led into a curious error. He commences thus, p. 31 : “ As the combination of two literal symbols in the form xy expresses the whole of that class of objects to which the names or qualities represented by X and y are together applicable, it follows that, if the two symbols have exactly the same signification, their combination expresses no more than either of the symbols taken alone would do. That is, wx=x+=X. The law which this expresses is practically exemplified in language. To say good good,' in relation to any subject, though a cumbrous and useless pleonasm, is the same as to say good.' Thus, good good’ men is equivalent to 'good' men.” Only two symbols of number obey this formal law. They are 0 and 1. Their natural interpretation in the system of logic is Nothing and Universe, which are the two limits of class extension. If from the conception of the universe, as consisting of " men ” and “ not-men,' we exclude or subtract the conception of “ men,” the resulting conception is that of the contrary class, "not-men.” Hence, if x represent mei), the class “not-men” will be represented by 1-X. And, in general, whatever class of objects is represented by the symbol x, the contrary class will bə. expressed by 1–x, which we may write a Boole now goes on to make the blunder referred to, in gravely stating : “ Prop. IV. That axiom of metaphysicians which is termed the principle of contradiction, and which affirms that it is impossible for any being to possess a quality and at the same time not to possess it, is a consequence of the fundamental law of thought, whose expression is æ+=.” As Mr. Venn has remarked, this “ surely argues a strange inversion of order.” Indeed, the inversion is so palpable that we are astonished to find Liard
repeating the error on page 292 of his article, where he says, “Maintenant il est aisé de voir que l'axiome appelé par les logicians principe de contradiction, et considéré par eux comme une loi primitive et irreductible de la pensée, est une conséquence de cette loi antérieure dont l'expression est: x=x.”
But while the law '=x should have been introduced as rather the effect than the cause of the principle of contradiction, yet I believe I am announcing an important discovery when I say that it is this law alone which has, so far, rendered division impossible in the algebra of logic, which in turn forced Boole to introduce the machinery and all the features which have been objected to in his calculus. I may add, in passing, that, having traced the difficulty to its source, I believe myself able to overcome it, and hope to publish my solution at no distant day.
To return to our author, he says, p. 36: “ Suppose it true that those members of a class, x, which possess a certain property, z, are identical with those members of a class, y, which possess the same property, z; it does not follow that the members of the class x universally are identical with the members of the class y. Hence it cannot be inferred from the equation zx=zy that the equation x=y is also true. In other words, the axiom of algebraists, that both sides of an equation may be divided by the same quantity, has no formal equivalent here.” He attempts no explanation of this anomaly, but makes it analogous to the case where, in the algebra of number, if, in the equation zx=zy, z can be 0, we cannot deduce x=y. Now, this is an eminently false analogy, only representing the case where z is the limiting class “ nought,” which, combined with any class, gives nought. Here the two algebras are completely analogous, but this is not at all the point we are considering. The special limitation in logical algebra is not caused by any one special class, like 0, but applies to every class and to all equations, and has nothing in the slightest degree analogous to it in the algebra of number. When he reverts to this matter again, p. 88, we see more conclusively that he has been able to think of no logical cause for it, and can only fall back on this false quantitative analogy. He says : “ If the fraction has common factors in its numerator and
é-e denominator, we are not permitted to reject them, unless they are mere numerical constants. For the symbols. x, y, etc., regarded as quantitative, may admit of such values, 0 and 1, as to cause the common factors to become equal to 0, in which case the algebraic rule of reduction fails. This is the case contemplated in our remarks on the failure of the algebraic axiom of division,' p. 36. Now, if there was no cause for the failure of the division axiom except the reduction of some factor to nought, there would be no cause for calling attention to the matter, and we might proceed to use division precisely as we do when treating of number, since a zero has precisely the same effect in both algebras.
But, in point of fact, Boole cannot use real division at all. If he chooses to write ayz=xz in the form a=*, he has not divided out any factor, and dare not. Even when he is certain z is not nought he cannot divide it out, which demonstrates instantly the falsity of his analogy. The real cause is the existence of the law, xx=x=x, in the logical algebra, which has no counterpart in that of number. That this is the true explanation appears very simply, as follows: If we have an equation in which a common factor appears in both members, as, e. g., zy=zx, this law renders it impossible for us to know how far the class z coincides with 2, since it may run from absolute difference up to complete identity; so that, in dividing out z, we may always be leaving some or all of it behind in the remaining factor. For example, if all rational white men = all white rational animals, and we divide out “ rational,” we have, all white men = all white animals. Now, the fact that this is not true, that a white man is not a white horse, though both are white animals, does not depend upon anything becoming zero, but upon the fact that on one side some of the meaning of rational has been unavoidably left behind in the term “men,” while the division succeeded in taking it all out of the other member of the equation. If we start with the simple truth, “ All men are all the rational animals,'' that is, m=ra, we may multiply both sides by r and it remains just as true; becoming rm=rra=ra=ra ... rm=ra. But, if now we attempt to divide out this r we just put in, it draws with it the original r from one member, while leaving it latent in the other member, and we have m=a, all men are all the animals.
This shows is why in Boole's system we cannot divide; and when, remembering this restriction, we use the fractional form, we get expressions which often bear on their face no meaning or interpretation. These Boole transforms, by what he calls development, into forms always strictly interpretable. The fact of his conducting his reasoning thus, through mediate uninterpretable steps, has been the most serious objection to his system, yet he saw no other way to attain a perfectly general solution.
This development theorem, given on p. 73, Prop. II, “ To expand or develop a function involving any number of logical symbols,” contains, and has been made, the basis of Stanley Jevons' whole logical system. Utterly misconceiving his master's attempt to give a genuine algebra of logic, which should make it a progressive science like quantitative mathematics, Mr. Jevons has been entirely content with the general method of indirect inference by trials, which is given immediately by this one theorem of development. We cannot enter here into a discussion of the principles involved in this process of generalized dichotomy. Merely as a hint at its application, we treat the simple proposition we have been using, m=ra. To get at what this can tell us about animals we express a as a function of m and r: a=". Developing, we have a="=
f (m.r)= f (1,1) m.rtf (1,0) m.rtf (0,1) mr+ f (0,0) mr. From this, without trials, Boole proves that all animals consist of all men and some irrational things not men. But, if he would have consented to use trials in referring to the premises in every particular instance, he would not have needed the co-efficients of his expansion. Thus, since all men are rational, the second term, m.7, strikes out; and, since men are all the rational animals, the third term, m.1, strikes out, and we are left for animals only mr and mr, as before. This