AND ON THE LOGIC OF RELATIONS. 335; 



If Newton were the examiner of my failures, I could recall the occasion on which he 

 lost his own connexion between the inverse square and the ellipse, because his casual 

 diagram put conjugate diameters at right angles to one another, and seduced him into 

 the belief that they were the principal axes. Were it Wallis, I could revive the time 

 when he hesitated at Jli = 2^3, sure of the theorem, but doubtful of the validity of 

 the expression, for want of precedent. Were it Leibnitz, I could bring to his memory 

 the co-inventor of the differential calculus, doubting whether to say yes or no to the 



equation — = d [-] , and working out the decision on paper. And so on. 



The want of power which most persons feel in the treatment of combined relations, 

 may be well illustrated by cases of the class of relationships which have almost appro- 

 priated the name, those of consanguinity and affinity. Many educated persons, and 

 some acute logicians, would either pause for an unreasonable time, or would not give 

 the right answer, if asked for all the conclusion* that follows about John and Thomas 

 from ' William is not John's father, and Thomas is William's uncle.' 



The only relations admitted into logic, down to the present time, are those which 

 can be signified by is and denied by is not. Allowing to the substantive verb all its 

 range of meaning — and that range is a wide one — and introducing contrary notions, 

 all the relations which were styled onymatic in my last paper, whether arithmetical, 

 mathematical, or metaphysical, are capable of inclusion. All other relation is avoided by the 

 dictum that it shall be of the form of thought to consider the relation and the related pre- 

 dicate as the predicate, and the judgment as a declaration or denial of identity between this 

 and the related subject. 



Accordingly, all logical relation is affirmed to be reducible to identity, A is A, to non- 

 contradiction. Nothing both A and not-A, and to excluded middle, Everything either A or 

 not-A. These three principles, it is affirmed, dictate all the forms of inference, and evolve all 

 the canons of syllogism. I am not prepared to deny the truth of either of these propositions, 

 at least when A is not self-contradictory, but I cannot see how, alone, they are competent to 

 the functions assigned. I see that they distinguish truth from falsehood : but I do not see 

 that they, again alone, either distinguish or evolve one truth from another. Every trans- 

 gression of these laws is an invalid inference: every valid inference is not a transgression of 

 these laws. But I cannot admit that every thing which is not a transgression of these laws 

 is a valid inference. And I cannot make out how just the only propositions which are true 

 of all things conceivable can he or lead to any distinction between one thing and another. 

 I believe these three principles to be of the soil, and not of the seed, though the seed may 



• The old riddle-books often propound the following 

 query: — If Dick's father be Tom's son, what relation is Dick 

 to Tom ? When a boy, I heard the following classical and 

 Protestant version of the puzzle, over which I have since made 

 grown persons ponder, not always with success. An abbess 

 observed that an elderly nun was often visited by a young 

 gentleman, and asked what relation he was. " A very near 

 relation," answered the nun ; " his mother was my mother's 



only child :" which answer, as was intended, satisfied the ab- 

 bess that the visitor must be within the unprohibited degrees, 

 without giving precise information. When this is proposed, 

 the first answer often is, He was her grandchild; and if 

 the story did not say that the visitor was young, he would 

 sometimes be taken for her grandfather; the matter not 

 preventing, <^^-' might as well be mistaken into <t>~^(p~^&i 

 into <^0. 



Vol. X. Part II. 43 



