SEPTEMBER 15, 1899. ] 
in the field of logic. The character of the 
work done by the two men is very different. 
De Morgan’s work bristles with new sym- 
bols ; Boole uses only the familiar symbols 
of analysis. The former polished many 
small stones ; the latter raised an edifice of 
grand proportions. The work done by Boole 
in applying mathematical analysis to logic 
necessarily led him to consider the general 
question of how reasoning is accomplished 
by means of symbols. The view which he 
adopted on this point is stated on page 68 
of the ‘ Laws of Thought.’ 
“The conditions of valid reasoning by 
the aid of symbols are: First, that a fixed 
interpretation be assigned to the symbols 
employed in the expression of the data, 
and that the laws of the combination of 
these symbols be correctly determined from 
that interpretation ; Second, that the formal 
processes of solution or demonstration be 
conducted throughout in obedience to all 
the laws determined as above, without re- 
gard to the question of the interpretability 
of the particular results obtained; Third, 
that the final result be interpretable in 
form, and that it be actually interpreted in 
accordance with that system of interpreta- 
tion which has been employed in the ex- 
pression of the data.”’ 
As regards these conditions it may be ob- 
served that they incline toward the realist 
view of analysis. True, he speaks of inter- 
pretation instead of meaning, but it is a 
fixed interpretation ; and the rules for the 
processes of solution are not to be chosen 
arbitrarily, but are to be found out from 
the particular system of interpretation of 
the symbols. The thoroughgoing realist 
view is that a symbol stands for some 
definite notion in the subject analyzed, and 
that the rules of the analysis are founded 
upon universal properties of the subject an- 
alyzed. The realist view of mathematical 
science has commended itself to me ever 
since I made an exact analysis of relation- 
SCIENCE. 
ool 
ship and devised a calculus which provides 
a notation for any relationship ; can express 
in the form of an equation the relationship 
existing between any two persons, and pro- 
vides rules by means of which a single equa- 
tion may be transformed, or a number of 
equations combined so as to yield any rela- 
tionship involved in their being true simul- 
taneously. The notation is made to fit the 
subject, and the rules for manipulation are 
derived from universal physiological laws 
and the more arbitrary laws of marriage. 
The basis is real ; yet the analysis has all 
the characteristics of a calculus, and throws 
light by comparison on several points in or- 
dinary algebra. Its fundamental symbol 
expresses a relation ; and what is the ulti- 
mate meaning of the algebraical symbol or 
of the symbol of the calculus but an opera- 
tion or relation ? 
It is Boole’s second condition which prin- 
cipally calls for study and examination ; re- 
specting it he observes as follows: ‘‘ The 
principle in question may be considered as 
resting upon a general law of the mind, the 
knowledge of which is not given to us 
a priori, i. e., antecedently to experience, but 
is derived, like the knowledge of the other 
laws of the mind, from the clear manifesta- 
tion of the general principle in the partic- 
ular instance. <A single example of reason- 
ing, in which symbols are employed in 
obedience to laws founded upon their inter- 
pretation, but without any sustained refer- 
ence to that interpretation, the chain of 
demonstration conducting us through inter- 
mediate steps which are not interpretable to 
a final result which is interpretable, seems 
not only to establish the validity of the par- 
ticular application, but to make known to 
us the general law manifested therein. No 
accumulation of instances can properly add 
weight to such evidence. It may furnish 
us with clearer conceptions of that common 
element of truth upon which the application 
of the principle depends, and so prepare the 
