Published in the Cambridge
and Dublin Mathematical Journal
Vol. III (1848), pp. 18398
Excerpt:
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...
