436 Mh DE morgan, on the syllogism, No. V. AND 



and all in two equivalent propositions, none in one and all in the other, use them both in 

 one and the same proposition. I have pointed out, in the fourth of the letters alluded to, 

 how an insufficient summary (IX. ii. 281) probably led Hamilton into the erroneous language 

 of the hurried article (VI. 6S5*) : it is hardly worth repeating here. 



I proceed to the further consideration of the system before us. I shall apply my own 

 notation to Hamilton's forms: thus X(-(y will designate 'Some X is not any Y\ It 

 will be seen that, in the doubly partitive system, no one proposition simply contradicts another: 

 though )•( and (•) would have done it if (•) had been truly brought under definition. 



I shall take for granted that when any premises are given, every conclusion which those 

 premises can yield must be drawn. I do not mean that in the common syllogism I must be 

 noted as a conclusion whenever A is so : because I can be inferred from A. I mean that 

 every possible conclusion must be stated, either immediately or mediately. I will grant 

 to the framer of a system the right to be governed by the hypotheses on which he sets out, 

 in the acceptance or rejection of any premises. But, should he accept a certain pair of 

 premises, I will not grant him the right to stifle a part of the conclusion because he has no 

 form in his system by which to express it: he ought to invent the form. Against any one who 

 demands such a right I quote Hamilton, who insists upon it that language is to be found 

 for all that is in thought : and I aver that when premises are put into the head, all the 

 conclusion is in thought to all who can master it. There are two ways of offending against 

 the reasonable principle stated above. Firs^, by curtailing the conclusion to as much as 

 can be expressed in the system. Secondly, by excluding combinations of premises because 

 they have no conclusion except what cannot be expressed in the system, and for no other 

 reason. Both these faults are committed : to which must be added the still greater fault of 

 conclusions which do not follow from the premises. 



The canon of validity laid down is that one premise must be affirmative (or both) ; 

 and that one middle term must be universal (or both). I take this from the earlier 

 writings, and by induction from the latest list of syllogisms : I shall not stop to consider 

 the general canon (IX. ii. 285). It will be remembered that by affirmative and negative 

 Hamilton refers to his own division, to his affirmatives which (all but one) contain negations, 

 and to those negatives which are but disjunctively joined affirmatives. Speaking his lan- 

 guage, and especially remembering that all his propositions are simply convertible, I affirm 

 that both articles of his canon of validity are erroneous. As follows : 



1. Both premises may be negative. Let us try )•()•). If 'Any X is not any Y' 

 and ' Any Y is not some Z', it follows that ' Some Z is not any Y', and the remaining Z is 

 Y, and therefore not X. Consequently, we have a right to the Aristotelian conclusion, 

 ' Some Z ia not X'. 



2. Both middle terms may be particular. Let us try ) ) ( ). If ' All X be some Y ' 

 and 'Some Y is some Z', whence ' Some Z (the rest) is not any Y', it follows that all this 

 remaining Z is not any X. Hence we have the Aristotelian conclusion, • Some Z is not X', 



Here we see pairs of premises yielding conclusions from which we are debarred, because 

 those conclusions are not such as require the doubly partitive 'some' to express them. I now 

 pass on to the syllogisms which are allowed admission (IX. ii. 287). 



