BS
The Calculus of Logic
George Boole
Cambridge and Dublin Mathematical Journal
Vol. III (1848), pp. 183-98
In a work lately published I have exhibited the application of a new and peculiar form of Mathematics to the expression of the operations of the mind in reasoning. In the present essay I design to offer such an account of a portion of this treatise as may furnish a correct view of the nature of the system developed. I shall endeavour to state distinctly those positions in which its characteristic distinctions consist, and shall offer a more particular illustration of some features which are less prominently displayed in the original work. The part of the system to which I shall confine my observations is that which treats of categorical propositions, and the positions which, under this limitation, I design to illustrate, are the following:
(1) That the business of Logic is with the relations of classes, and with the modes in which the mind contemplates those relations.
(2) That antecedently to our recognition of the existence of propositions, there are laws to which the conception of a class is subject, - laws which are dependent upon the constitution of the intellect, and which determine the character and form of the reasoning process.
(3) That those laws are capable of mathematical expression, and that they thus constitute the basis of an interpretable calculus.
(4) That those laws are, furthermore, such, that all equations which are formed in subjection to them, even though expressed under functional signs, admit of perfect solution, so that every problem in logic can be solved by reference to a general theorem.
(5) That the forms under which propositions are actually exhibited, in accordance with the principles of this calculus, are analogous with those of a philosophical language.
(6) That although the symbols of the calculus do not depend for their interpretation upon the idea of quantity, they nevertheless, in their particular application to syllogism, conduct us to the quantitative conditions of inference.
It is specially of the two last of these positions that I here desire to offer illustration, they having been but partially exemplified in the work referred to. Other points will, however, be made the subjects of incidental discussion. It will be necessary to premise the following notation.
The universe of conceivable objects is represented by 1 or unity. This I assume as the primary and subject conception. All subordinate conceptions of class are understood to be formed from it by limitation, according to the following scheme.
Suppose that we have the conception of any group of objects consisting of Xs, Ys, and others, and that x, which we shall call an elective symbol, represents the mental operation of selecting from that group all the Xs which it contains, or of fixing the attention upon the Xs to the exclusion of all which are not Xs, y the mental operation of selecting the Ys, and so on; then, 1 or the universe being the subject conception, we shall have
x 1 or x = the class X,
y 1 or y = the class Y,
xy 1 or xy = the class each member of which is both X and Y,
and so on.
In like manner we shall have
1 - x = the class not-X,
1 - y = the class not-Y,
x(1 - y) = the class whose members are Xs but not-Ys,
(1 - x)(1 - y) = the class whose members are neither Xs nor Ys,
&c.
Furthermore, from consideration of the nature of the mental operation involved, it will appear that the following laws are satisfied.
Representing by x, y, z any elective symbols whatever,
x(y+z) = xy + xz . . . . . . . .(1),
xy = yx, &c. . . . . . . . .(2),
x = x, &c. . . . . . . . .(3).
From the first of these it is seen that elective symbols are distributive in their operation; from the second that they are commutative. The third I have termed the index law; it is peculiar to elective symbols. |
