THE THEORY OF SYLLOGISM, ETC. 



KC5 



One however may be mentioned, as giving the mode of arrangement preferred by Sir 

 William Hamilton. The preceding is a table of double entry, and the passage from compart- 

 ment to compartment in the same column shews that all the negative syllogisms are formed 

 from the affirmative ones by simple insertion of the negative distinction into one or the other 

 premise and the conclusion. Thus, in the second column, at the head of the three compart- 

 ments, stand )))( = )( and ))).( = ).( and ).)) ( = ).(. In the cumular system, the modified 

 modes of inference occur with certain affirmative propositions, which are therefore exceptional. 



Looking at the preceding as one way of meeting the formal difficulties of Sir William 

 Hamilton's system, it will easily be conceived possible that there may* be others, for one or 

 more of which we are to look to its inventor, and one more of which I shall presently give 

 myself. But, in the manner in which it has been given out, up to the present time, the defects 

 which I have pointed out exist unanswered. 



The exemplar system and that of contraries have 21 syllogisms in common. Of the re- 

 maining 15 of the first, 7 correspond to 7 of the syllogisms with both premises negative in the 

 second. The remaining 8 of the first belong, a pair to each of the four syllogisms of the 

 second which have both middle terms particular, one of them being the remaining syllogism 

 with premises both negative. When a symbol taken from the exemplar system is invalid if 

 read in the other, first try it by annexing the negative sign to each premise which is affirmative : 

 if this do not make it valid, alter the curvature of both the middle parentheses, and then either 

 the addition or removal of a negative symbol will make it valid. 



Section V. 



ON THE THEORY OF THE COPULA AND ITS CONNEXION WITH THE DOCTRINE OF FIGURE. 



The analysis of the so called simple proposition, or judgment, shews that it is, in every 

 one of its particulars, of a complex character. This is readily admitted in regard to the terms, 

 and to the quantity in its connexion with them. But the copula has always been considered 

 as the extreme both of simplicity and generality ; and any attempt at the resolution of the 

 copular relation into its elements, is likely to be misunderstood. 



In algebra, as it now stands, the forms born and educated in arithmetic have left their 

 parent and set up for themselves. Any meanings which obey certain specified laws may be 

 adopted as the means of giving expression to the forms : and the results must be accepted as 



* Of course I might instance my own numerically definite 

 system, which, as Sir William Hamilton acknowledges, con- 

 tains his system, and which is not open to any charge of incon. 

 sistency. The acknowledgment was made in the assertion 

 that my numerical system, so far as it gives expressed quantity 

 of all kinds to the predicate, was derived from information 

 furnished by him. I am still waiting for two citations : one 



from Sir William Hamilton's communication, containing in- 

 formation ; one from my own subsequent writings, containing 

 use made of it. I want, in fact, an exemplar proposition of 

 the form )( : having hitherto obtained nothing but ) ), or ( ) : 

 I have had enough of single indefiniteness, and want an in- 

 stance of the double character which gives singularity to the 

 things to be compared. 



