THE CALCULUS OF LOGIC

In a work lately published[1],I have exhibited the application of a new and peculiar form ofMathematics to the expression of the operations of the mind inreasoning. In the present essay I design to offer such an account ofa portion of this treatise as may furnish a correct view of thenature of the system developed. I shall endeavour to state distinctlythose positions in which its characteristic distinctions consist, andshall offer a more particular illustration of some features which areless prominently displayed in the (p. 184)[2] original work. The part of thesystem to which I shall confine my observations is that which treatsof categorical propositions, and the positions which, under thislimitation, I design to illustrate, are the following:

(1) That the business of Logic is with the relations of classes, andwith the modes in which the mind contemplates those relations.

(2) That antecedently to our recognition of the existence ofpropositions, there are laws to which the conception of a class issubject,—laws which are dependent upon the constitution of theintellect, and which determine the character and form of thereasoning process.

(3) That those laws are capable of mathematical expression, and thatthey thus constitute the basis of an interpretable calculus.

(4) That those laws are, furthermore, such, that all equationswhich are formed in subjection to them, even though expressed underfunctional signs, admit of perfect solution, so that every problem inlogic can be solved by reference to a general theorem.

(5) That the forms under which propositions are actually exhibited, inaccordance with the principles of this calculus, are analogous withthose of a philosophical language.

(6) That although the symbols of the calculus do not depend for theirinterpretation upon the idea of quantity, they nevertheless, in theirparticular application to syllogism, conduct us to the quantitativeconditions of inference.

It is specially of the two last of these positions that I here desireto offer illustration, they having been but partially exemplified inthe work referred to. Other points will, however, be made thesubjects of incidental discussion. It will be necessary to premisethe following notation.

The universe of conceivable objects is represented by 1 or unity.This I assume as the primary and subject conception. All subordinateconceptions of class are understood to be formed from it bylimitation, according to the following scheme.

Suppose that we have the conception of any group of objects consistingof s s, and others, and that ...