AND ON THE LOGIC OF RELATIONS. 



345 



language has some degree of tendency towards the same sort of enlargement of meanings. 

 As in the very example before us : the brothers of the other parent are called brothers (in 

 law): and under this extension uncle of child is identical with brother: for the word uncle 

 receives similar extensions. 



A relation is said to be convertible (though it should rather be said that the subject and 

 predicate are convertible) when it is its own converse; when X..LY gives Y..LX. And, 

 L being any relation whatever, LL~' is convertible: but LL"'' and L^L"' are each the 

 converse of the other. So far as I can see, every convertible relation can be reduced to the 

 form LL~'. If two notions stand in the same relation to one another, they can always, I 

 tliink, be made to stand in one and the same relation to some third notion. The converse 

 is certainly true, namely, that two notions which stand in one relation to a third, stand in con- 

 vertible relation to each other. But it cannot be proved that if X..LY and Y..LX, then 

 L must be reducible to MM"', for some meaning or other of M : this is certainly a material 

 proposition. But I can find no case in which material proof fails. Take identity, for ex- 

 ample : it is the very notion of identity between X and Y that X..LL"'Y for every possible 

 relation L in which X can stand to any third notion. Identification of objects of thought 

 by names derives its convertibility from the idea of the names standing in relation of appli- 

 cability to the same object. Identification in thought of unnamed objects can only be con- 

 ceived as convertible by reference, as above, to other notions. Exclude names, and identify 

 X with itself by ' this is this :' it would be absurd to repeat the process, and say that there 

 is conversion by reason of the first this of one indication being the second this of the other: 

 such conversion would be only the invention of different names spelt the same way. 



Among the subjects of a convertible relation must usually come the predicate itself, 

 unless it be forced out by express convention. If all convertible relation can be expressed 

 by LL~' this is obviously necessary: for LL~iX includes X. Is a man his own brother? 

 It is commonly not so held: but we cannot make a definition which shall by its own power 

 exclude him, unless under a clause expressly framed for the purpose. Is a brother the son'*' 

 of the same father and mother with the man himself? Then is he pre-eminently his own 

 brother : for there never lived one of whom we have not more reason to be sure that he was 

 the son of his own father and mother than that any reputed brother had the same parents 

 with him. If we want to exclude him we must stipulate that by brother of X we mean any 

 man who, not being X himself, has the same father and mother. In common language the 

 stipulation is or is not made, according to the casual presence or absence of the necessity for 

 it. Put the question what relation to a man is his brother's brother, and most persons will 

 answer, His brother : point out that the answer should be, Either his brother or himself, and 



* When the individual is but one among many, and is 

 speaking generally of his class, there is an implication that 

 others are intended, and the introduction of self produces for 

 a moment that sense of incongruity which, if it could be made 

 to last, would give an air of humour. Thus Hobbes, in a 

 sentence which, altering geomelris into logicis, might be said 

 of himself by a person I ought to know, speaks as follows : 



In magna quidem periculo versari video emstimationem meam, 

 qui a geomelris fere omnibus dissentio. Eorum enim qui de 

 iisdem rebus meeum aliquid ediderunt, aut solut insanio ego 

 out solus non insanio ; terlium enim non est, nisi (quod dicet 

 forte uliquis) insaniamus crmnes. Undoubtedly a man is 

 among those who have written on the same subjects with him- 

 self. 



44—2 



