October 20, 1892] 



NATURE 



587 



each other. Therefore, the angles at the base of any (or every 

 or all) isosceles triangle are equal to each other. 



In order to make the nature of this reasoning plainer, I will 

 put the same in symbols. 



Put X = the isosceles triangle in general ; 

 Z = this particular isosceles *riangle ; 

 a = angles at the base ; 

 e — equal to each other. 

 Then our reasoning will appear thus : — 

 Z is X, 

 <z of Z is ^ ; 

 . • . rt of X is ^. 

 This looks very much as though we had in hand a case of the 

 logic of relatives. 



We will all recollect the challenge of De Morgan : — " If any 

 one will by ordinary syllogism prove that because every man is 

 an animal, therefore every head of a man is a head of an animal, 



I shall be ready to set him another question." This would 



be in symbols — 



All M is A ; 

 .-. AofM is /i of A. 

 Our case, according to this sort of formulation, would appear — 

 Z is any X ; , 



.'. a of Z is a of X, 

 « of Z is ^ ; 

 . • . a of X is *. 

 The distinguishing characteristic of our case as compared with 

 the case put by De Morgan resides in the different natures of the 

 two propositions — 



All M is A, 

 and Z is any X. 



The former is the usual universal categorical affirmative pro- 

 position of ordinary logic. The latter is a sort of universal cate- 

 gorical affirmative proposition that certainly exists and is im- 

 portant, but which has not yet been recognized, unless it may 

 be by the quantifiers of the predicate in their proposition — 



All A is all B. 

 It implies rigorously that not only 



Z is any X, 

 but that Z is every X, 



Z is all X, 

 any (or every or all) X is Z j 



,, ( ,, ,, ) not X is not Z 



,. ( „ „ ) ,. Z „ X I 



any X is every Z j 



every X is any Z j 



every not X is any not Z | 



any ,,,,,, \ 



&c., &C. j 



In truth, to a superficial notice, it may easily seem to confuse the 

 most important logical distinctions. But this is only because 

 we are so used to identifying logic in general with the logic of 

 extension. It is the logic of extension, or in other words, ! 

 metric logic, that has been persistently tendered to us as the only 

 logic worthy of study, if not mdeed as the only logic practicable 

 or perhaps possible. Yet we can now, I think, see that [ 

 geometry at least makes great use of a logic that is not the 

 logic of extension, and that the existence of geometry is an ' 

 earnest that that other logic may be developed and formulated, I 

 if not completely at least to some very useful extent. 

 The proposition, 



Z is any X, 



is, as I conceive, a proposition of the logic of intension. It 

 applies not to things, or to concepts in connection with things, 

 but to pure abstract concepts like geometrical figures, whose 

 marks are exhaustively specified, or, if any are not specified, 

 the same depend upon and are implied by those that are 

 specified. 



One point more remains to be explained. 



We must not from the recognition that Z is any X, and the 

 rigorously following proposition that whatever is true of either 

 X or Z is true of the other, conclude that these propositions 

 should if valid hold for marks that are accidental to Z, or to 

 any single instance of X. If we fail to keep clearly in mind the 

 intensive scope of our propositions, we may discredit them, or 



one of them, by observing tbat although we have laid down that 

 whatever is true of Z is true of any X, yet neverthelci-s it does 

 not follow that Z is any X, as regards say the size of Z or any 

 single instance of X. Size may not be any nece>sary mark of 

 either, and if so it is for all logical purposes impertinent to the 

 propositions in question, and must be altogether ignored. 



I will conclude by saying that the inference actually made in 

 the case put by Miss Jones is a deduction, because it necessarily 

 follows from the premises laid down. Logic has no connection 

 with the truth of premises ; it only says what certain proposi- 

 tions entail. If in an intensive sense this isosceles triangle is 

 any isosceles triangle, then any isosceles triangle is this isosceles 

 triangle, and every isosceles triangle is this isosceles triangle, 

 and all isosceles triangles are this isosceles triangle, and if the 

 angles at the base of this isosceles triangle are equal to each 

 other, it follows necessarily that the angles at the base of all 

 isosceles triangles are equal to each other. 



Chicago, August i6. • Francis C. Rus-ell. 



By the courtesy of the Editor of Nature I have been allowed 

 to read Mr. Francis C. Russell's very interesting remarks with 

 reference to my note on "Induction and Deduction " in Nature 

 of July 28 (p. 293). 



I agree with Mr. Russell as to the validity and certainty of an 

 inference from equality of angles in one isosceles triangle tc 

 equality of angles in a/l isosceles triangles. But while I' re- 

 gard this inference as a "true induction " because it is an in- 

 ference to a general proposition on the strength of a particular 

 instance, Mr. Russell denies that it is an induction because he 

 holds that induction can give only approximate and probable 

 conclusions, and considers that the certainty which he allows 

 to belong to the geometrical conclusion in question is due 

 to the fact that the inference is noi from a particular in- 

 stance, but is really and truly from universal /o universal— the 

 one isosceles triangle from which the argument starts being a 

 kind of " pure abstract concept," so that we can say — 



7'Ats one isosceles triangle is any isosceles triangle ; there- 

 fore every isosceles triangle is this isosceles triangle, &c. 



This appears to me to be entirely inadmissible. How can 

 this triangle BE that and the other triangle ? To say that it is, 

 is to lose sight of the distinction between identity of indi- 

 viduality, and similarity of characteristics. And that the asser- 

 tion {this triangle is every triangle) is untenable appears also 

 from Mr. Russell's own admission further on, when he says that 

 "we must not, from the recognition that Z is any X, and the 

 rigorously following proposition, that whatever is true of X or 

 Z is true of the other, conclude that these propositions should, 

 if valid, hold for marks that are accidental to Z or to any single 

 instance of X." If Z is any X, how can any X have marks 

 which Z has not, or Z have marks which any X has not ? We 

 cannot get out of the difficulty by reference to extension and 

 intension, for this reason, that every categorical proposition, to 

 be significant, must be read both in "intension" and in "ex- 

 tension" — that is, affirmatives must be understood as asserting 

 identity of extension (application) in diversity of intension 

 (signification), while negatives dtny identity. " This isosceles 

 triangle is any isosceles triangle " can have a useful signification 

 only if it is interpreted to mean — 



This triangle [not is but] is siviilar (in so far as isosceles) to 

 any isosceles triangle— that is, all are similar in respect of the 

 characteristics which are inseparable from equality of sides. 

 Hence (as I said in my letter, July 28) "in all cases equality of 

 angles at the base is inseparable from equality of sides." 



1 am not clear what precise meaning can be attached to the 

 expression "pure abstract concept," still less how a geometrical 

 figure can be an abstract concept. I am, moreover, disappointed 

 that Mr. Russell makes no examination whatever of my own 

 attempt to formulate the process from Particular to General. 



With reference to Mr. Russell's symbolical argument — 



ZisX, 



a of Z is e, 

 . • . a of X is ^, 



I think that it may be logically described as a process either 

 (i) of Substitution (Jevons) — a kind of Immediate Inference 

 dependent on identity of application — thus : — 



Z is X ; 



, •. X may be substituted for Z. 



NO. II 99, VOL. 46] 



