AND ON LOGIC IN GENERAL. 



225 



word former referring to my first paper on syllogism : — " In the former I have expressed the 

 quantity of my conclusion, there called the middle term, being as much as is really middle, by 

 m + n — 1 ..." Mv critic made what I call a wrong* hyphenism : he read it " quantity of ray 

 conclusion-there-called-/Ae-middle-terra " instead of quantity-of-my-conclusion, there called the 

 middle term." If two leaves had been turned back, so that the extracts from " the former" 

 should have met the eye, there would have been seen — " If these fractions be m and n, then 

 the middle term is at least the fraction m + n - \ of the Ys." The meaning is clear. If the 

 fraction tn of the Ys be Xs, and the fraction n of the Ys be Zs, then the extent ' the fraction 

 m + n — 1 of the Ys' is really in both terms, and is " as much as is really middle.'"'' I defend 

 the grammar of my phrase, though not its sufficient completeness : but it is well enough for an 

 allusion to matter of four pages back. In ' the A of my B, there called C,' C refers to A, 

 unless context make it very apparent that the writer is not correct. If I had wanted to refer 

 C to B, I should have said, 'the A of my B, which B is there called C 



• I should have left the cause of this wrong hyphenism to 

 the conjecture of my reader if I had not, while this paper was 

 in proof, noticed the juxtaposition and opposition of the two 

 passages given below. I have long been satisfied that I knew 

 why a metaphysician who, right or wrong, was seldom weak, 

 proclaimed the quantitative sciences of space and time to be 

 mental nuisances; but, except for the passages now given, I 

 could not have pointed to the explanation without much colla- 



" This being understood, the Table at once exhibits the real 

 identity and rational differences of Breadth and Depth, which, 

 though denominated quantities, are, in reality, one and the 

 same quantity, viewed in counter relations and from opposite 

 ends. Nothing is the one which is not, pro tanto, the other." 



Of two quantities, then, which are "one and the same 

 quantity," the less the one, the greater ihe other ! The writer 

 of these passages never failed to convey anything he really 

 had to convey, by want of power over language: when the 

 Bank ran short, be knew the way to the Mint. But he was 

 here suffering under an insufficiency of notion. He wanted 

 to grasp the idea that of his two quantities one determines the 

 other. The conception of functional relation was struggling 

 in a mind whicli could not furnisli it with a locus standi for 

 want of an adequate conception of quantity. A little further 

 on he proceeds thus : — " Though different in the order of 

 thought {ratione), the two quantities are identical in the nature 

 of things (re). Kach supposes tiie other; and Breadth is not 

 more to be distinguished from Depth, than the relations of the 

 sides, from the relations of the angles, of a triangle." Had 

 "identical" here had its meaning, it would have been absurd 

 to say that Breadth and Depth are "not more" distinct than 

 the relations of the sides and the relations of the angles of a 

 triangle : it ought to have been " not so much by a great deal." 

 But, as it is, and choice of word apart, my eminent critic is 

 right: Breadth and Depth have relations each to the other. 

 The word " identical," as used by him, synibolises his aspira- 

 tion after the notion of mutual dependence, which mathema- 

 ticians call functional relation. He has pushed to an extreme 

 a liberty with the word identical which is not uncommon 

 among logicians. They start from "X is X" as the expression 

 of identity ; and their extensions of the verb involve exten- 



Vol. X. Paut I. 



tiori and laboured inference. I hope the reader will verify for 

 himself, first, the following quotations, next, my assertion that 

 the context would not reconcile them in the slightest degree. 

 In both editions of the Discussions, within eight lines of each 

 other, and as parts of the same train of thought, occur the fol- 

 lowing sentences (l»t. cd. p. fi44*; 2nd. ed. p. 699). The 

 "table" and the " diagram" are one. 



" The two quantities are thus, as the diagram represents, 

 precisely the inverse of each other. The greater the Breadth, 

 the less the Depth ; the greater the Depth, the less the Breadth ; 

 and each, within itself, affording the correlative difl'erences of 

 whole and part, each, therefore, in opposite respects, contains 

 and is contained,^^ 



sions of the substantive. But they do not handle simultaneous 

 extensions of cognate words with as much facility as the ma- 

 thematicians, to whom the process is of constant occurrence, 

 sometimes forced upon them by the progress of their science, 

 sometimes excogitated, pro re nata, to forward that progress. 



Quantity, in the mind of my critic, was the res divisibilis 

 (§ xxi); and not correctly conceived even in this sense. We 

 may now see how it comes to be affirmed, in the dissertation I 

 have quoted, that equation of quantities is convertible with 

 coalescence of notions, and equation with identification: that 

 a competent notation must "sublimate into one" identical 

 quantities: that though a proposition be "merely" equation 

 of quantities, — two Breadths, or two Depths, — it may, at will, 

 be considered in neither quantity. No one of these assertions 

 wants truth, provided that clear meaning be given by reduction 

 of "quantity" within its true limits, and restoration of "coa- 

 lescence" to its proper place. Other writings of the same 

 author contain phrases illustrative of the theory on which I 

 account for those alre«dy produced. Such as that moods are 

 numerically equal in all figures ; meaning that all figures have 

 the same numbers of moods : such again as the phrase amplifi- 

 cation of number; meaning the addition of other number to 

 it : such again as addition to the number of things described 

 as increase in the things themselves. And lastly, quantity of 

 the conclusion not merely confounded with tlie conclusion 

 itself, but the confusion attributed to a mathematician. 

 I {June 25, 1858.) 



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