94 



PROFESSOR DE MORGAN, ON THE SYMBOLS OF LOGIC, 



But in truth this basis is excessive, in three distinct ways. 



1. (((( and )))) really belong to one of the sets which is to be formed; and the pro- 

 ceeds of the two will be identical. One only of them should be taken. 



2. The sets from ())) and ((() only differ in respect to the species of figure* which 

 I employ. I keep to XY, YZ, XZ; and one of these sets is the other referred to ZY, 

 YX, ZX : the refusal of all distinction of figure obliges me to retain both sets. 



3. In (()) = () we see only a strengthened syllogism : that is, stronger in the premises 

 than is required to produce the conclusion. It is either ( ))) or ((( ) with the particular 

 middle term universalized. 



The number of variations produced by combining X or x, Y or y, Z or z, is eight, taking 

 one out of each pair. And as these variations do not affect the character of the proposition, we 

 must have as follows. Eight universal syllogisms, with two universal premises and a universal 

 conclusion, derived from )))) = )). Sixteen particular syllogisms, each with a particular 

 conclusion, derived from one universal and one particular premise ; eight commencing with 

 a particular, from ())) = (); eight commencing with a universal, from ((( ). Eight strength- 

 ened syllogisms, in which two universals conclude with a particular, derived from (()) = ( ). 



The canon of validity is as follows. The change of Y into y will alter the middle paren- 

 theses by changing them both, and therefore will not affect their relative character. Conse- 

 quently, if the middle parentheses turn the same way, ((or )), any two propositions which give 

 this arrangement allow of an inference, if one at least be universal ; that is, if the number of 

 oppositions in the parentheses, and the number of negative dots, put together, be an odd num- 

 ber in all, or an even number (0 included) in each proposition. But when the middle parentheses 

 turn contrary ways, there need but be two universals, or an even number (0 included) on each 

 side. 



The canon of inference is merely this ; — Strike out the middle parentheses, and two 

 negative dots, if there be two ; the remaining symbol shews the inference. 



* I thought I had sufficiently expressed this in my work by 

 stating that 1 abandoned the distinction of figure. But I sup- 

 pose it was hardly clear: for a learned reviewer observes, 

 "That is not new — but is Sir William Hamilton's avowed 

 rule." This is not correct : Sir William Hamilton's method is 

 equally indifferent to all figures ; mine holds by one, and 



recognizes no other, in its classifications. I have one establish- 

 ment, and tolerate no distinction of sect. Sir William Hamil- 

 ton has none, but tolerates three sects : Aristotle had three 

 state-churches, Galen (they say) founded a sect of dissenters, 

 which, after some centuries of toleration, was made a fourth 

 establishment. 



