TRANSACTIONS OF THE SECTIONS. 13 



mentioned some with the remark that they do not depend upon the dictum dc 

 omni et nullo, but on something of equivalent evidence. The only question ia 

 whether we should be right in considering these cases as exceptions, and if they 

 are so, to what they owe their existence. One instance is the inversio relationis, 

 e. y., Noah is Shem's father, therefore Shem is Noah's son. Here we pass from the 

 idea of Shem to that of his father, and vice versd. The movement of the mind is 

 along a track distinct from that which it follows, either in algebra or what we com- 

 monly call logic. The perception of the truth of the inference depends on a recog- 

 nition of the correlation of the two ideas, father and son. Again, take a similar 

 instance. Prince Albert sat at the Emperor's right hand, therefore the Emperor 

 eat at Prince Albert's left hand. How shall we express such inferences symboli- 

 cally ? Let S be Shem, N Noah,/ father, s son : 



N=/S, 

 S/=l. 



Eliminating/ between these two equations, we get 



S=sN. 



Nothing can be simpler than this ; but the symbols s, / are of a distinct nature 

 from those employed in the * Laws of Thought ; ' for /A does not denote a 

 species of A, but an idea standing in a different relation to it. The distinction 

 between these two kinds of symbols becomes more manifest when we reflect that 

 /^ is not identical with /, but denotes "father of father," or grandfather. Now I 

 do not see how these cases of inversion of relation are to be dealt with symbolically 

 without the introduction of such symbols. In the following examples I confine 

 myself to the cases afforded by relationship, and the succession of generations. 



Let A, B, C denote three persons, s son, g gi-andson ; then if J3 is A's son and 

 C is B's, C is A's grandson, which we may express symbolically by the following 

 equations : — 



'B=sA, C=sB, s^=ff. 

 Eliminating B and s, we get C=//A. It would be more accurate in these examples 

 to introduce a symbol x or y to indicate that B is only one of the possible sons of 

 A, an individual ranged under the species sA. I shall do that in the next example, 

 in which the word son is replaced by the more general term descendant, denoted 

 by d. The equations will now be 



B=.r(?A, C =yf/B, d^=zd, 

 viz. a descendant not of the first generation. The result of eliminating B now is, 



C=yd.vdA ; 



but by a principle about to be noticed dx=x'd, therefore C — px'zdA, or C is in- 

 cluded in the class of descendants of A. 



The principle just used forms one of the recognized examples of an inference not 

 lying within the domain of Aristotelian logic. It was called " transitio ex recto in 

 ohliquum." Whately, though he says nothing of its nature, gives in his praxis of 

 examples one which depends upon it. A negro is a man, therefore he who kills a 

 negro kills a man. Let this derived notion killing be denoted by/, which may 

 serve to indicate a general functional dependence, then Mand N denoting man and 

 negro respectively, we have the following equations : — 



N=:rM,/r = :r/; 

 .•./N = .r/M, 

 or the killing of a negro is a kind of homicide. The evidence of the truth of the 

 equation /rs=.r/ is the same as that in favour of the equation xy=yx, when .rand 2/ 

 both belong to the kind of symbols used in the ' Laws of Thought.' I sliall not stop 

 to inquire into the limitations which it may perhaps require. 



The general truth of the equations 



x'^=x and xy=yx 

 appears to suffer another exception in the case relative terms, that is, of adjectives 

 of which the interpretation is functional of the object to which they are applied. 

 A small St. Bernard dog is not simpliciter a small dog; the word meaning that 

 which is less than the medium size of the class of objects to which it is applied. 



