OF ARTS AND SCIENCES : MARCH 12, 1867. 251 



and 



(4.) ( a _| r j)_[ rC _ a _| r( &_|_ c) . 



Let a , b denote the individuals contained at once under the classes 

 a and b ; those of which a and b are the common species. If a and b 

 were independent events, a , b would denote the event whose proba- 

 bility is the product of the probabilities of each. On the strength of 

 this analogy, (to speak of no other,) the operation indicated by the 

 comma may be called logical multiplication. It is plain that 



(5.) a ,a == a. 



Logical multiplication is evidently a commutative and associative 

 process. That is, 



(6.) a ,b== b ,a 



(7.) (a,b) ,c = a,(b,c). 



Logical addition and logical multiplication are doubly distributive, 

 so that 



(8.) (a-\rb),c== a,c -\rb,c 



and 



(9.) o,5 4-c=5=(a-|rc),(J-|rc). 



Proof. Let a=za'-\-x-\-y-\-o 



b ===. V -\- x -\- z-\- o 



where any of these letters may vanish. These formulas compre- 

 hend every possible relation of a , b and c ; and it follows from them, 

 that 



a-\rb==a' -f J' -\- x -\- y -\- z -\- o (a -\r b) , c === y -f- z -f- o. 



But 



0^ = ^ + b,c = z-\-o a,c-{rb,c=py-{-z-\-o .-.(8). 



So 



a ,b == x -f- o a,b-\rC==c'-\-x-{-y-\-z-\-o. 

 But 



(a-\rc)=a'-{-&-\-x-}-y+z+o (&-}rc)z=5' + c'+a:+y + «+ 

 {a-\rc) , (b + c) == & + x +y + z + o .: (9). 



