The Calculus of Logic — George Boole
The calculus of logic | Project Gutenberg You may copy it, give it away or re-use it under the terms of the Project Gutenberg License included with this eBook or online at www.gutenberg.org . If you are not located in the United States, you will have to check the laws of the country where you are located before using this eBook. Title : The calculus of logic Author : George Boole Release date : December 10, 2022 [eBook #69512] Most recently updated: April 1, 2023 Language : English Original publication : United Kingdom: MACMILLAN, BARCLAY, AND MACMILLAN, 1848 Other information and formats : www.gutenberg.org/ebooks/69512 Credits : Laura Natal Rodrigues and Ray Papworth (Images generously made available by The Internet Archive.) *** START OF III (1848), pp. 183-98 In a work lately published [1] , 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 (p. 184) [2] 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 s s, and others, and that , which we shall call an elective symbol, represents the mental operation of selecting from that group all the s which it contains, or of fixing the attention upon the s to the exclusion of all which are not s, the mental operation of selecting the s, and so on; then, 1 or the universe being the subject conception, we shall have and so on. In like manner we shall have Furthermore, from consideration of the nature of the mental operation involved, it will appear that the following laws are satisfied. Representing by , , any elective symbols whatever, 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. The truth of these laws does not at all depend upon the nature, or the number, or the mutual relations, of the individuals included in the different classes. There may be but one individual in a class, or there may be a thousand. There may be individuals common to different classes, or the classes may be mutually exclusive. All elective symbols are distributive, and commutative, and all elective symbols satisfy the law expressed by (3). These laws are in fact embodied in every spoken or written language. The equivalence of the expressions "good wise man" and "wise good man," is not a mere truism, but an assertion of the law of commutation exhibited in (2). And there are similar illustrations of the other laws. With these laws there is connected a general axiom. We have seen that algebraic operations performed with elective symbols represent mental processes. Thus the connexion of two symbols by the sign + represents the aggregation of two classes into a single class, the connexion of two symbols as in multiplication, represents the mental operation of selecting from a class those members which belong also to another class , and so on. By such operations the conception of a class is modified. But beside this the mind has the power of perceiving relations of equality among classes. The axiom in question, then, is that if a relation of equality is perceived between two classes, that relation remains unaffected when both subjects are equally modified by the operations above described . (A). This axiom, and not "Aristotle's dictum," is the real foundation of all reasoning, the form and character of the process being, however, determined by the three laws already stated. It is not only true that every elective symbol representing a class satisfies the index law (3), but it may be rigorously demonstrated that any combination of elect