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.

The expression All Ys represents the class Y and will therefore be expressed by y, the copula are by the sign =, the indefinite term, Xs, is equivalent to Some Xs. It is a convention of language, that the word Some is expressed in the subject, but not in the predicate of a proposition. The term Some Xs will be expressed by vx, in which v is an elective symbol appropriate to a class V, some members of which are Xs, but which is in other respects arbitrary. Thus the proposition A will be expressed by the equation

y = vx

On Syllogism.

The forms of categorical propositions already deduced are 

• y = vx, All Ys are Xs,

•  y = v(1 − x), No Ys are Xs, 

• vy = v'x, Some Ys are Xs, 

• vy = v'(1 − x), Some Ys are not-Xs, 

whereof the two first give, by solution, 1 −x = v' (1−y). All not-Xs are not-Ys, x = v' (1 − y), No Xs are Ys. To the above scheme, which is that of Aristotle, we might annex the four categorical propositions

• 1 − y = vx, All not-Ys are Xs, 

• 1 − y = v(1 − x), All not-Ys are not-Xs, 

• v(1 − y) = v'x, Some not-Ys are Xs, 

• v(1 − y) = v'(1 − x), Some not-Ys are not-Xs,

the two first of which are similarly convertible into 

• 1 − x = v'y, All not-Xs are Ys,

• x = v'y, All Xs are Ys,  or No not-Xs are Ys, 

If now the two premises of any syllogism are expressed by equations of the above forms, the elimination of the common symbol y will lead us to an equation expressive of the conclusion.

Ex. 1.

• All Ys are Xs, y = vx,

• All Zs are Ys, z = v'y,

the elimination of y gives 

• z = vv'x,

the interpretation of which is

• All Zs are Xs,

the form of the coefficient vv' indicates that the predicate of the conclusion is limited by both the conditions which separately limit the predicates of the premises.