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enclosure of M ; B is an enclosure of M ; therefore A and B 
are co-enclosures of M ; ” or, by dropping the M, “ A and B 
are co-enclosures.” As an instance of this in expressing 
actual relations : — “The Duke of Wellington was an Irish- 
man; Lord Palmerston was an Irishman; therefore the 
Duke of Wellington and Lord Palmerston were fellow- 
irishmen ; ” or, by dropping the name of the particular 
nation, “The Duke of Wellington and Lord Palmerston 
were fellow-countrymen.” 
The relation of inclusion is here expressed by L, and 
its converse by L~^. Co-inclusion, or the relation of one 
enclosure to another is expressed by ^ or X® ; co-includence, 
jj—i 
or the relation of one includent to another by or 
In this system, so long as the contrary is not stated, 
enclosure always means enclosure in the same includent, 
and includence means includence of the same enclosure. 
Consequently from the premises “A and M are co-includents; 
M and B are co-includents ; ” we have the conclusion that 
“ A and B are co-includents.” In the common logic, on the 
contrary, the premises “ Some A is M ; some M is B,” yield 
no conclusion. 
The fundamental canon of this system is that the enclo- 
sure of an enclosure is an enclosure ; and conversely, that 
the includent of an includent is an includent. These are 
expressed by the equations L^—L and The 
former of these is the canon of the old “Syllogism in 
Barbara ; ” the latter is the same read backwards. These 
equations express that the relation of inclusion and inclu- 
dence is transitive. 
The relations of co-enclosure and co-includence are also 
transitive, and they are, moreover, invertible, — that is to 
say, if A=X®B or A=(X~^)®B, then B=X®A or B==(X“^)°A. 
These are expressed by the equations and 
To express this in words: — every term 
