84 PROFESSOR DE MORGAN, ON THE SYMBOLS OF LOGIC, 



volved. When I say "'John met Thomas in the street' — 'John shook hands with Thomas' — 

 that is, 'John shook hands with what John met in the street'" — there is an elimination of 

 ' Thomas' perfectly answering in process to 'a + = b, y = c + x, therefore y = c + b - a'. 



Several remarkable matters connected with inference may be made to arise out of this view, 

 of which some will be noticed. But I now proceed to observe that this perfect sameness of 

 logical and algebraical process does not continue. Whenever close resemblances exist which 

 are rarely or never noted, we may be pretty sure that the attention has been diverted by 

 differences as remarkable or more so. In algebra, in which the result is quantity, equations 

 are perfect identities. If x be 10, every x that can be produced under precisely the same 

 circumstances, is entitled to the copular sign (=) in connexion with any abstract 10 that we can 

 imagine. The consequence is, that a complete conversion of all the processes of elimination 

 can always be made. If y = <px, % = \j/x give * = %y, then * = \j/x and % = yjy always give 

 y = <hx as one at least of certain alternatives. From y = c + x, y = c + b — a, we can recover 

 a + x = b, with which we began : but from 'John shook hands with Thomas' and 'John shook 

 hands with what he met in the street', we cannot of right recover 'John met Thomas in the 

 street.' 



When an assertion becomes complicated, we may be, as in algebra, in the position of finding 

 the reduction impossible : or else we may make the beginner's mistake of disentangling the 

 subject (solving the equation) by the process of ignotum per ignotum. For instance 'John 

 met Thomas near his own house:' if we describe John as 'a man who met Thomas near his 

 own house,' we may not have a solution, for the pronoun is logically the noun. But we have 

 one if we refer 'his own 1 to ' a man.'' 



The following is an imitation of an elimination of two quantities between three equations : 



John met Thomas ; John and Thomas live in the same street ; 



John is richer than Thomas. The elimination gives either of three forms, of which this 

 is one ; the algebraic process is here fully imitated. 



' A person who met one not so rich as him whom he met' and ' that same person not 

 so rich as him whom he met' live in the same street. 



These assimilations may appear ludicrous, but it will be presently seen that the ideas 

 which they suggest may help to free logic from being to the higher processes of thought what 

 algebra would have been to its present state if it had discussed no forms more complicated 

 than x = y, y = *, &c. The world at large is more advanced in constructive process than 

 the books on logic ; in which arbitrary separations are declared by old authority to con- 

 tain all the forms of thought. 



There are many processes of inference which are not syllogisms nor reducible to them, and 

 they all belong to a form of elimination which is not precisely that of common algebra. In 

 this last science we use identities because we can command identities. We have not much 

 occasion to employ x>y or x<y, because, from our experience of identities, we find it more 

 convenient to suppose x = y - b, or x = y + b ; making the indeterminate character of b supply 

 the want of precision which appears in x>y, as compared with x = y. Nevertheless, elimination 

 between inequalities is sometimes required: and then we know that in x>y our right of sub- 

 stitution is, that we may for x write an equal or a greater, for y an equal or a less. In x>y, 



