January 21, 1892] 



NATURE 



273 



economize your space, I only gave the skeleton of the argu- 

 ment, but I hoped I had said enough to indicate at least the 

 general outline of my logical views. But as this seems not to 

 have been quite the case, may I now explain myself a little more 

 fully ? 



I may remove a slight misunderstanding at once. I said our 

 knowledge of our own continuous existence in the present is to 

 each of us a necessary truth. Dr. Mivart reads this as if I had 

 written "our continued existence in the future"! That we 

 cannot be annihilated while we know that we are existing is, 

 as I shall presently show, not a mere consequence of the law of 

 contradiction. If this law is of any use at all in proving the 

 conclusion, it would certainly be useless without a second pre- 

 miss, viz. that we are existing ; and this latter is the premiss 

 which is a necessary truth. 



I suppose everyone will acknowledge that a definition is 

 essentially an arbitrary assertion, and that therefore a definition 

 can by itself give no real information. But a well-understood 

 term does not consist of a definition alone. Its definition may 

 be laid down, as a list of items of connotation (or denotation), 

 and the other part of its meaning, which may be called its 

 import, that is its denotation (or connotation) must be dis- 

 covered by experience ; and the knowledge so acquired is real, 

 not only verbal, knowledge. Now it is possible from a number of 

 definitions alone to deduce a series of propositions. These, like 

 the definitions from which they were deduced, give by them- 

 selves only verbal information — they are all truisms — and before 

 they can be made of any practical use, certain real assertions, 

 assigning real import to the terms, and so expressing real know- 

 ledge, must be added to the premisses. Thus, if we wish to 

 determine whether any given proposition is a truism, or conveys 

 real information, we have only to examine the definitions of its 

 terms. If these are found to be inconsistent with each other, 

 the proposition is a contradiction in terms, and must be re- 

 jected. If the definitions are not inconsistent, but are inde- 

 pendent of each other, the proposition can only be intended to 

 assert the identity of the import of its terms — it therefore con- 

 veys real information, which may either be true or false. Lastly, 

 if the definitions can be shown to be dependent on each other, 

 the proposition is equally true whatever import its terms may 

 have, or even if they have no conceivable import at all. It is a 

 truism. If, however, by the aid of other real propositions any 

 real import can be given to its terms, it may have objective, or 

 subjective, applications ; but the objectivity or subjectivity is 

 introduced by those other propositions, and is not a property of 

 the original truism. 



Take, for example, the proposition, " Everything must either 

 * be ' or ' not be '" ; or the proposition, "Twice two is four." 

 The truth of either of these propositions depends solely on the 

 definitions of its terms, as I pointed out in my last letter, and 

 this is why I cannot regard them as objective truths. Of course 

 I do not doubt that if I had lost an eye I should not remain in 

 the same condition as I was before. But, although "no man 

 out of Bedlam would suppose a statement of a general law 

 would inform us about a concrete thing," this is precisely what 

 Dr. Mivart does if he regards the above proposition as depend- 

 ent solely on the law of contradiction. Does he not see that he 

 added the objective element to that law in the phrase, "if he 

 had lost an eye " ? "Much virtue in If." The status of the 

 proposition, "Two straight lines cannot inclose a space," 

 similarly depends on the definitions of its terms ; but, as I 

 pointed out in my last letter, these terms may be defined in two 

 different ways — either by dependent definitions, so making the 

 proposition a truism, or independently so as to make it a real 

 assertion, in which case it might conceivably be false. Dr. 

 Mivart apparently takes the former set of definitions, and then 

 implies that I deduced the latter result from them, which, if he 

 reads my letter again, he will find not to have been the case. 



In reply to Miss Jones, I may point out — 



(i) It most certainly is merely a verbal convention when Miss 

 Jones says, " A and a ' are not applicable to the same thing.' " 

 She had herself just before laid down the convention in ques- 

 tion, in the phrase, " If A signifies the negation of a (whatever 

 A may stand for)." 



I do not know why Miss Jones should imagine that I think 

 that " assertions (or denials) of the * existence ' of particular 

 objects are the only real propositions," but perhaps she will 

 understand my view better when she has read this letter. 



(2) I certainly hold that " inductions have no logical justifica- 

 tion whatever," if by "logic" is to be understood formal, or, 



NO. 1160, VOL. 45] 



as I prefer to call it, symbolic, reasoning. The essence of in- 

 duction, in my opinion, is the assumption (at first arbitrarily) of 

 an hypothesis to account for observed facts— that is, ultimately, 

 of directly apprehended sensations. The full significance of the 

 hypothesis is elucidated by symbolic reasoning, and the enumer- 

 alio simplex is applied to the results of this rea-^oning. and does 

 not, therefore, appear quite in the simple form exhibited by Miss 

 Jones. But it remains equally true that no induction can ever 

 lead to a necessary truth. 



(3) Miss Jones's view of mathematical reasoning is exactly 

 that which I wish to combat. We do not, in mathematics, 

 conclude a universal proposition from a single concrete instance, 

 A mathematical formula does not imply the existence of any 

 instance whatever of its application, any more than a definition 

 implies the reality of the thing defined. The formula is deduced 

 from what may logically be regarded as definitions, and one or 

 any number of applications may indeed be found afterwards, 

 but only by the aid of additional real premisses. It is difficult 

 to exemplify this in the case of geometry, because the accepted 

 geometrical methods are so very imperfect, and geometrical 

 conclusions are not always deduced from definitions alone. As 

 I implied in my former letter, some of them are founded on 

 induction. But it must be evident that the truth of, say, De 

 Moivre's Theorem, does not depend on our having seen that it 

 was true in any one instance. 



(4) If Miss Jones reads her own paragraph (4) again carefully, 

 I think she will see that it is not I who have contradicted 

 myself. I showed that if the definitions of the terms of a certain 

 proposition were altered, the proposition might no longer be 

 true, and that if they were not altered it would always be true. 

 Argal, the truth depen4s on the definitions, and on nothing else. 



1 did not maintain that it could ever be to anyone a necessary 

 truth that he was writing with a lead-pencil. That would be an 

 objective proposition, such as I was careful to insist could only 

 be proved by induction. It might, however, be a necessary 

 truth to anyone that he thought he was writing with a lead- 

 pencil. As to mathematical truths, so far from believing that 

 "in as far as ' real ' they are obtained by induction," I expressed 

 my opini n that they are not "real" at all, but all truisms. 

 Any reality in their applications must be added from outside, by 

 real assertions which are not " mathematical." I object to call- 

 ing truisms "necessary," not because they are possibly false, 

 but because their truth is only arbitrary. On the other hand, 

 when I call "the apprehension of a present fact" a necessary 

 truth, I mean something more than that it is certain — namely, 

 that its contradictory is unthinkable. 



Edward T. Dixon. 



Trinity College, Cambridge, January 8. 



FRESH EVIDENCE CONCERNING THE DIS- 

 TRIBUTION OF ARCTIC PLANTS DURING 

 THE GLACIAL EPOCH. 



LAST summer (1891) I spent some weeks in Western 

 Russia and Northern Germany, in order to ascer- 

 tain whether the glacial fresh-water deposits of those 

 countries contained any remains of the vegetation which 

 lived there immediately after the inland ice had melted 

 away. The results of my journey being favourable, I 

 have thought it desirable to communicate them to the 

 readers of Nature ; but before doing so it might be con- 

 venient to give a brief summary of previous investiga- 

 tions on the same subject. 



The first discovery of fossil Arctic plants was made in 

 England by Mr. W. Pengelly, who found in i860, at 

 Bovey Tracey, in Devonshire, leaves of the dwarf birch 

 {Betula nana), together with leaves of some willows, 

 as Salix myrtilloides, S. cinerea, S. sp. indet. The 

 leaves were identified and described by the late Prof. 

 Heer,^ who pronounced the opinion that the presence ot 

 Betula nana was conclusive evidence of "a colder 

 climate than Devonshire has at the present day." The 

 significance of this discovery was, however, but little 

 appreciated until the researches mentioned below again 



' Philosophical Transactions. 1862, p. 1039 In this paper Heer mentions 

 Salix repens (?>, but this determination "was subsequently altered to S. 

 tnyrtilloides. 



