AND ON THE LOGIC OF RELATIONS. 



349 



knowledge, in this instance, converts the conclusion into ' X is neither the brother of Z, nor 

 the brother of all the wives who have had children by Z.' Again, let the conclusion be, X is 

 not any N of Z. If then X be N of Z, and Z parent of Y, X must be uncle of Y: that is 

 ' N of parent ' is ' uncle,' and N is to be found by decomposition. What of parent is uncle ? 

 The aggregate of brother and brother-in-law. But to say that X is neither brother nor 

 brother-in-law would be to suppose that Y might be any child of Z : so that all we are to 

 say is, X is neither brother, nor brother-in-law with reference to one particular child. Drop 

 the vestige of the middle term, and we say that X is neither brother nor brother-in-law by 

 every wife : which agrees with the preceding. 



The mode of decomposition may be thus generalised. Let there be one negative premise, 

 and, L and M being the premising relations, let N be the concluding relation denied. 

 Write down the terms of the negative premise, and between them the remaining term, 

 choosing such order as shall make X precede Z. Combine the relations of the two 

 pairs seen, and the combination must be the direct or converted relation in the negative 

 premise, provided that due attention be paid to the particular character in the affirmative 

 premise. For example, let the negative premise be Z . MY. Take Y, Z, and between them 

 write X, YXZ ; in which YX, XZ, are seen. Let the affirmative premise be X .. LY: 

 then L~' and N combine to give M~'; or N must be deduced by making L~'N identical 

 with M~^, a certain L"' being understood. 



I have had occasion to notice the manner in which, by wilful renunciation of knowledge, 

 the conclusion is made to express not quite all the possible inference. This happens also 

 in the common syllogism. If from ' Some Xs are not Ys' and ' Everything is either Y or Z,' 

 we deduce ' Some Xs are 2s' it is not true that this conclusion embraces the whole knowledge 

 which the premises give. It is known that ' some Xs' mean all that are not Ys : the vague 

 quantity is not so vague as it would be if the conclusion were the only thing known. It ought 

 to be noticed that a universal (Hamilton's definite would be a better word) lurks in the 

 conclusion of every particular syllogism : in the above X () Z would be Xy )) Z, if all 

 that is known were expressed. The particular conclusion of a syllogism is the universal of 

 a narrower name : one premise predicates existence for a new and compounded name : the 

 conclusion substitutes that compound in the other premise in a legitimate manner. 



Reserving the word mood as irrevocably associated with details of quantity and quality 

 combined, let each figure have four phases, determined by the quality only of the premises. 

 The four phases, + meaning affirmative, and — meaning negative, are to be remembered* in 



the standard order , , , , 



+ + — I- H • 



such persons : it is material that if Z be either Whiston or 

 Dr Primrose, the brother of the wife would be sufficient. All 

 this and more is conceded : matter makes its appearance the 

 moment L and M mean more than ' certain relations which, 

 and no others, are designated by these letters throughout the 

 syllogism.' Again, when I am told that Logic does not pro- 

 vide the inference that ' Philip was Alexander's father ' 

 because ' Alexander was Philip's son,' and that it is our 

 material knowledge of the relation of father and son that 

 enables us to make the inference, I reply that it is certainly 

 material that father and son are related in the manner of h and 



L-'; but that the transition from X..LY to V..L-'X is a 

 form of thought, and a more general form than any case of 

 conversion admitted by the logician in the common syllogism. 

 It is that which is common to the transitions ' X a genus of 

 Y, therefore Y a species of X ', ' X a parent of Y, therefore 

 Y a child of X,' ' X an identical of Y, therefore Y an iden- 

 tical of X,' &c. : and is therefore more abstract than any of 

 them, and equally form without matter to all of them. 



• Some persons, of whom I am one, but whether it be a 

 gift or a defect I do not know, cannot associate two things 

 with two other things, each with each, merely by conventional 



