66 
MR. A. J3. EEMPE ON THE THEORY OF MATHEMATICAL FORM. 
370. When any collection £b, £c, £cl, ... of classes of a full set T have certain 
relations of inclusion, &c., to each other; instead of saying that £a, £b, £c, £d, . . . have 
these relations, we may say that a, b, c, d, . . . have the relations “ in A class £ 
may be completely defined by stating that it is the most extensive class in which 
certain relations hold between other classes. Thus the class a-\-b of a full set in 
which a and b are unrestricted is fully defined by saying that it is the most extensive 
class of the set in which “ all a is b,” or in which ab=a. Further, instead of using 
this definition, we may simply employ the equation ab — a without other words; it 
will then assert that the class ci-\-b is that under consideration and will be a mark or 
term denoting that class. The equation a—u will be a term denoting the class a of 
the set. 
371. We might use ab—a to denote, not the class a-\-b, but the class which 
contains all those classes arrived at by ignoring the relations of inclusion, &c., 
between a-\-b and the various classes contained in a+6.* It is in this sense that 
such a mark is usually employed. 
372. Taking ab—a as used in the last paragraph, the term ab-f^a, i.e., “some a 
only is b,” will represent the class arrived at by ignoring the relations of inclusion, 
&c., between all those classes of T which are not included in ab=a. Here ab=f=a 
is the obverse of ab — a in P the full set derived from the full set T by regarding the 
classes of the latter as separated. 
373. Terms such as ah—a, ab^a, are called propositions. The difference between 
propositions and other terms is accidental and not essential matter of exact thought; 
propositions, like other terms, merely denote classes. 
374. When we consider a full set T of 2" classes ra, rb, tc, rd, ... , and the 
<p)l 
related propositional classes, we consider a system S of 2 3 ~ classes, viz., that derived 
from the 2 s unrestricted classes a, b, c, d,. . . ; the classes r and tt being other classes 
of S. If as is generally the case T is regarded as derived from n unrestricted classes, 
represented by single letters A, B, C, D, . . . , the number of classes in T will be 
2 2 ", in P will be 2 s2 , in S will be 2 r . 
375. In a syllogism we have two classes called the premisses, e.g., (1) (ab=ai), 
(2) ( bc = b ). We consider the class which is the product of these, viz., 
(ab—a) ( bc = b), i.e., that which contains all the separated classes of T, which both 
(1) and (2) do. This class includes some classes, is included in others, &c. It is 
included in ( ac=a ). The syllogism indicates this fact, the class ( ac=a ) being 
called the conclusion. 
376. We have then under discussion in an investigation of classes a system of 
2 ra units consisting of 
* If the proposition “ all a is l ” is taken to imply tlie existence of its subject a, we must exclude 
the zero class of T. Similarly in the case of the proposition “some a only is b ” (sec. 372). 
