9 
r 
represent either qualities, or the classes defined by the 
qualities ; and the combination of two or more such symbols 
represents either the combination of the qualities, or the 
class defined by such combination. The combination of 
these symbols is the logical analogue of multiplication in 
common Algebra. The inverse of multiplication is division, 
and the inverse of logical combination is abstraction. 
''Let us express the proposition, Man is the rational 
animal, by 
m = ra. 
To every multiplication (of two unlike terms) correspond 
two divisions, which here give 
m T m 
— = r and — = a, 
a r ’ 
whereof the meaning is that man without the animal at- 
tributes is a being of ^ure reason, and man without reason 
is a mere animal.”^ 
In Boole’s system, as in common algebra, it is generally 
true that 
lx = x, 
1 being the symbol for all. But his symbols are also sub- 
ject to a law unlike anything in common algebra, which is 
expressed by the equation 
XX ^x or x^=^ 'x, 
of which the interpretation may be thus stated in his own 
words : — “ The law which this expresses is practically ex- 
emplified in language. To say ' good good ’ in relation to 
any object, though a cumbrous and useless pleonasm, is the 
same as to say ' good.’ Thus ' good good ’ men is the same 
as ‘ good ’ men.” 
Under these laws of multiplication, what will be the 
* From my paper on a kindred subject, read before the Manchester Philoso- 
sophical Society on 11th December, 1883, 
