450 Mr DE morgan, ON THE SYLLOGISM, No. V. AND 



the possible ambiguity arising out of the dual and plural did or did not dictate adherence 

 to the singular ; the plural must have frequently taken in the dual, frequently not : that is, 

 the plural must have frequently meant two or more, frequently more than two. If this 

 be the explanation, it does not alter the fact. And to the fact must be joined the 

 utter extinction of the exemplar form in the minds of modern logicians : an extinction so 

 thorough that such a chart of logical history as Hamilton's mind had not that course laid 

 down, even as a possibility. It was easier to him to imagine Aristotle talking such 

 "nonsense" as that a form must be rejected because it was not true of all matter, than 

 meaning his singular number to speak of one: and a table of exemplar forms published 

 in 1850, appeared to him to stand alone in history. 



The exemplar form of statement is that both of geometry and of algebra. A proposition 

 in Euclid assumes some one case of satisfaction of hypothesis, and the demonstration lies 

 in the perception of the receiving mind that nothing in the reasoning is adverse to the implied 

 assertion that this some one may be any one. But this form of exact science is more pointedly 

 exemplar in phraseology than the system of Aristotle : its distributor is quilibet, not (minis. 

 We have an exemplar ladder in English : its steps are any one, each one, every one {quilibet, 

 unusquisque, omnis). The third is certainly not all ; for it is but one : but it is more truly the 

 grammatical singular of all than either of the others, near as they are. It would be difficult 

 to describe the differences of meaning : that there are differences will appear by our being able 

 to make sentences in which all three shall occur, without power of transposition. For ex- 

 ample : — " If you feel able to cope with any one, try each one, and so you will master 

 every one'''' — the order cannot here be altered. The first, any one, has a purer unitarian 

 character than the others : the second and third are more nearly transposible. Without 

 further inquiry, any one is the most proper for strict exemplar use, as being applicable in 

 negative predicates. 



The form 'any one X is any one Y' is much wanted in geometry. In my last paper 

 I pointed out that many indirect demonstrations are only refusals of the knowledge of 

 contraposition ; others, far less excusable, arise from refusal of the right to convert ' any one 

 X is any one Y.' When X and Y are of the same number of instances, the propositions 

 ' Every X is Y ' and ' Every Y is X ' are equivalent : which is most evident, if there be 

 any gradations of evidence, when X and Y are singular. Consequently, if there be but one 

 X and one Y, and if the X be the Y, it need not be proved that the Y is the X. Suppose 

 that a person, holding himself to have shown that Junius was an individual, and knowing 

 that Philip Francis was an individual, and that Francis was Junius, were to proceed as follows 

 to prove that Junius was Francis : — If not, let Junius be X, another than Francis : then be- 

 cause Francis is Junius, and Junius is X, it follows that Francis is X ; that is, Francis 

 is another than Francis, which is absurd. So it is, and so are you too, would be the answer 

 of common sense to the proposer of such a proof: is the principle of difference so much clearer 

 than that of identity, that any one has a right to suppose the sameness of ' X is Y ' and ' Y is X ' 

 to want corroboration by help of X being no other than itself.'' But this is done in geometry. 

 Not to insist on antiquity, let us take Legendre, a professed amender of Euclid : he knows that 

 through a point can be drawn but one perpendicular to a plane, and one parallel to a line ; yet 



