,,rtr99,,nd ,r99,,ss99,,ll 999999       
,,the ,,philosophy ,,of   ,,mr4         
,b99,,rtr99,,nd ,r99,,ss99,,ll    ,,with
 ,,an ,,appendix ,,of ,,leading         
,,passages ,,from ,,certain ,,other     
,,works    ,,edited ,,by    ,,philip ,e4
 ,b4 ,,jourdain    ,illustration        
,,london3 ,,george ,,allen  ,,unwin     
,,ltd4   ,,ruskin ,,house #dj ,,museum  
,,street1 ,w4,c4 #a   ,,chicago3 ,,the  
,,open ,,court ,,publishing ,,co4       
,first published in #aiah    7,all      
rights reserved7     ,,editor',s ,,note 
,when ,mr4 ,b99rtr99nd ,r99ss99ll1      
following the advice of ,mr4 ,w99ll9999m
 ,j99m99s1 again got into touch with    
reality and in ,july #aiaa was torn to  
pieces by ,anti-,suffragists1 many of   
whom were political opponents of ,mr4   
,r99ss99ll and held strong views on the 
,necessity of ,protection of ,trade and 
person1 a manuscript which was almost   
ready for the press was fortunately     
saved from the flames on the occasion   
when a body of eager champions of the   

,sacredness of ,personal ,property burnt
 the late ,mr4 ,r99ss99ll's house4 ,this
 manuscript1 together with some further 
fragments found in the late ,mr4        
,r99ss99ll's own interleaved copy of his
 ,prayer-,book of ,free ,man's ,worship1
 which was fortunately rescued with a   
few of the great author's other         
belongings1 was first given to the world
 in the ,monist for ,october #aiaa and  
,january #aiaf1 and has here been       
arranged and completed by some other    
hitherto undecipherable manuscripts4    
,the title of the above-mentioned       
,prayer-,book1 it may perhaps be        
mentioned1 was apparently suggested to  
,mr4 ,r99ss99ll by that of the ,essay on
 ,the ,free ,man's ,worship in the      
,philosophical ,essays 7,london1 #aiaj1 
pp4 #ei-gja7 of ,mr4 ,r99ss99ll's       
distinguished contemporary1 ,mr4        
,bertrand ,russell1 from whom much of   
,mr4 ,r99ss99ll's philosophy was        
derived4 ,and1 indeed1 the influence of 
,mr4 ,russell extended even beyond      

philosophical views to arrangement and  
literary style4 ,the method of          
arrangement of the present work seems to
 have been borrowed from ,mr4 ,russell's
 ,philosophy of ,leibniz of #aijj2 in   
the selection of subjects dealt with1   
,mr4 ,r99ss99ll seems to have been      
guided by ,mr4 ,russell's ,principles of
 ,mathematics of #aijc2 while ,mr4      
,r99ss99ll's literary style fortunately 
reminds us more of ,mr4 ,russell's later
 clear and charming subtleties than his 
earlier brilliant and no less subtle    
obscurities4 ,but1 on the other hand1   
some important points of ,mr4 ,russell's
 doctrine1 which first appeared in books
 published after ,mr4 ,r99ss99ll's      
death1 were anticipated in ,mr4         
,r99ss99ll's notes1 and these           
anticipations1 so interesting for future
 historians of philosophy1 have been    
provided by the editor with references  
to the later works of ,mr4 ,russell4    
,all editorial notes are enclosed in    
square brackets1 to indicate that they  

were not written by the late ,mr4       
,r99ss99ll4  ,at the present time we    
have come to take a calm view of the    
question so much debated seven years ago
 as to the legitimacy of logical        
arguments in political discussions4 ,no 
longer1 fortunately1 can that intense   
feeling be roused which then found      
expression in the famous cry1           
,justice--right or wrong1 and which     
played such a large part in the politics
 of that time4 ,thus it will not be out 
of place in this unimpassioned record of
 some of the truths and errors in the   
world to refer briefly to ,mr4          
,r99ss99ll's short and stormy career4   
,before he was torn to pieces1 he had   
been forbidden to lecture on philosophy 
or mathematics by some well-intentioned 
advocates of freedom in speech who      
thought that the cause of freedom might 
be endangered by allowing ,mr4          
,r99ss99ll to speak freely on points of 
logic1 on the grounds1 apparently1 that 
logic is both harmful and unnecessary   

and might be applied to politics unless 
strong measures were taken for its      
suppression4 ,on much the same grounds1 
his liberty was taken from him by those 
who remarked that1 if necessary1 they   
would die in defence of the sacred      
principle of liberty2 and it was in     
prison that the greater part of the     
present work was written4 ,shortly after
 his liberation1 which1 like all actions
 of public bodies1 was brought about by 
the combined honour and interests of    
those in authority1 occurred his        
lamentable death to which we have       
referred above4  ,mr4 ,r99ss99ll        
maintained that the chief use of        
implication in politics is to draw      
conclusions1 which are thought to be    
true1 and which are consequently false1 
from identical propositions1 and we can 
see these views expressed in ,chapters  
,,iii and ,,xix of the present work4    
,these chapters were apparently written 
before the ,government1 in the spring of
 #aiaj1 arrived at the famous secret    

decision that only certain implications 
are permitted in discussion4 ,naturally 
the secret decision gave rise to much   
speculation among logicians as to which 
kinds of implication were barred1 and   
,mr4 ,r99ss99ll and ,mr4 ,bertrand      
,russell had many arguments on the      
subject1 which naturally could not be   
published at the time4 ,however1 after  
,mr4 ,r99ss99ll's death1 successive     
prosecutions which were made by the     
,government at last made it quite clear 
that the opinion held by ,mr4 ,r99ss99ll
 was the correct one4 ,there had been   
numerous prosecutions of people who1    
from true but not identical premisses1  
had deduced true conclusions1 so that   
the possible legitimate forms of        
implication were reduced4 ,further1 the 
other doubtful cases were cleared up in 
course of time by the prosecution of    
7#a7 members of the ,aristotelian       
,society for deducing true conclusions  
from false premisses2 7#b7 members of   
the ,mind ,association for deducing     

false conclusions from false premisses2 
and also by the attempted prosecution of
 an eminent lady for deducing true      
conclusions from identities4            
,fortunately this lady was able to      
defend herself successfully by pleading 
that one eminent philosopher believed   
them to be true--which1 of course1 means
 that the conclusions are false4 ,thus  
appeared the true nature of legitimate  
political arguments4  ,,footnotes3  #a  
,this ,essay is also reprinted in ,mr4  
,russell's ,mysticism and ,logic1       
,london and ,new ,york1 #aiah1 pp4      
#df-eg4--;,,ed4         ,even a joke    
should have some meaning4444            
7,the ,red ,queen1 ,t4 ,l4 ,g41 p4      
#aje74     ,,contents                   
,,page   ,,editor',s ,,note             
#c   ,,abbreviations                    
#i   ,,chapter       ,i4 ,,the          
,,indefinables ,,of ,,logic             
#aa      ,,ii4 ,,objective ,,validity   
,,of ,,the ,,laws ,,of ,,thought        
#ae     ,,iii4 ,,identity               

#af      ,,iv4 ,,identity ,,of ,,classes
                                        
#ah       ,v4 ,,ethical ,,applications  
,,of ,,the ,,law ,,of ,,identity        
#ai      ,,vi4 ,,the ,,law ,,of         
,,contradiction ,,in ,,modern ,,logic   
#ba     ,,vii4 ,,symbolism ,,and        
,,meaning                               
#bb    ,,viii4 ,,nominalism             
#bd      ,,ix4 ,,ambiguity ,,and        
,,symbolic ,,logic                      
#bf       ,x4 ,,logical ,,addition ,,and
 ,,the ,,utility ,,of ,,symbolism       
#bg      ,,xi4 ,,criticism              
#bi     ,,xii4 ,,historical ,,criticism 
#cj    ,,xiii4 ,,is ,,the ,,mind ,,in   
,,the ,,head8                           
#ca     ,,xiv4 ,,the ,,pragmatist       
,,theory ,,of ,,truth                   
#cb      ,,xv4 ,,assertion              
#cd     ,,xvi4 ,,the ,,commutative ,,law
                                        
#ce    ,,xvii4 ,,universal ,,and        
,,particular ,,propositions             
#cf   ,,xviii4 ,,denial ,,of            

,,generality ,,and ,,generality ,,of    
,,denial            #cg     ,,xix4      
,,implication                           
#ci      ,,xx4 ,,dignity                
#dc     ,,xxi4 ,,the ,,synthetic        
,,nature ,,of ,,deduction               
#de    ,,xxii4 ,,the ,,mortality ,,of   
,,socrates                              
#dh     ,,xxiii4 ,,denoting             
#ec      ,,xxiv4 ,,the                  
#ed       ,,xxv4 ,,non-,,entity         
#ef      ,,xxvi4 ,,is                   
#eh     ,,xxvii4 ,,and ,,and ,,or       
#ei    ,,xxviii4 ,,the ,,conversion ,,of
 ,,relations                            
#fj      ,,xxix4 ,,previous             
,,philosophical ,,theories ,,of         
,,mathematics         #fa       ,,xxx4  
,,finite ,,and ,,infinite               
#fc      ,,xxxi4 ,,the ,,mathematical   
,,attainments ,,of ,,tristram ,,shandy  
#fd     ,,xxxii4 ,,the ,,hardships ,,of 
,a ,,man ,,with ,,an ,,unlimited        
,,income        #ff    ,,xxxiii4 ,,the  
,,relations ,,of ,,magnitude ,,of       

,,cardinal ,,numbers         #fi        
,,xxxiv4 ,,the ,,unknowable             
#gj      ,,xxxv4 ,,mr4 ,,spencer1 ,,the 
,,athanasian ,,creed1 ,,and ,,the       
,,articles    #gc     ,,xxxvi4 ,,the    
,,humour ,,of ,,mathematicians          
#gd    ,,xxxvii4 ,,the ,,paradoxes ,,of 
,,logic                                 
#ge   ,,xxxviii4 ,,modern ,,logic ,,and 
,,some ,,philosophical ,,arguments      
#gi     ,,xxxix4 ,,the ,,hierarchy ,,of 
,,jokes                                 
#ha        ,,xl4 ,,the ,,evidence ,,of  
,,geometrical ,,propositions            
#hc       ,,xli4 ,,absolute ,,and       
,,relative ,,position                   
#hd      ,,xlii4 ,,laughter             
#hf     ,,xliii4 ,,gedankenexperimente  
,,and ,,evolutionary ,,ethics           
#hh            ,,appendixes             
#hi     ,,abbreviations     ,a4 ,a4 ,w4 
,lewis ,carroll3 ,alice's ,adventures in
 ,wonderland1 ,london1                  
#aijh4 ,this book was first published   
much earlier1 but               this was

 the edition used by ,mr4 ,r99ss99ll4   
,the same applies               to ,h4  
,s4 and ,t4 ,l4 ,g4    ,a4 ,c4 ,p4      
,john ,henry ,blunt 7ed4 by73 ,the      
,annotated ,book of ,common             
,prayer1 ,london1 new edition1 #ahhh4   
,a4 d4 ,l4  ,ernst ,schroeder3          
,vorlesungen ueber die ,algebra der     
,logik1               ,leipzig1 vol4 i41
 #ahij2 vol4 ii4 7two parts71 #ahia and 
#aije2 vol4 iii43 ,algebra und ,logik   
der ,relative1 #ahie4    ,e4 ,n4        
,richard ,dedekind3 ,essays on the      
,theory of ,numbers1                    
,chicago and ,london1 #aija4    ,e4 ,l4 
,l4  ,william ,stanley ,jevons3         
,elementary ,lessons in ,logic1         
,deductive and ,inductive4 ,with copious
 ,questions and               ,examples1
 and a ,vocabulary of ,logical ,terms1  
,london1               #bdth ed41 #aijg 
first published in #ahgj4    ,e4 u4 ,i4 
,ernst ,mach3 ,erkenntnis und ,irrtum3  
,skizzen zur               ,psychologie 
der ,forschung1 ,leipzig1 #aijf4    ,f4 

,l4     ,augustus ,de ,morgan3 ,formal  
,logic3 or ,the ,calculus of            
,inference1 ,necessary and ,probable1   
,london1 #ahdg4    ,fm4 ,l4    ,john    
,neville ,keynes3 ,studies and          
,exercises in ,formal                   
,logic1 #dth ed41 ,london1 #aijf4       
,gg4       ,gottlob ,frege3             
,grundgesetze der ,arithmetik           
begriffschriftlich abgeleitet1 ,jena1   
vol4 i41 #ahic2               vol4 ii41 
#aijc4    ,gl4       ,gottlob ,frege3   
,die ,grundlagen der ,arithmetik3 eine  
logisch-mathematische ,untersuchung     
ueber den ,begriff der                  
,zahl1 ,breslau1 #ahhd4    ,g4 u4 ,e4   
,g4 ,heymans3 ,die ,gesetze und         
,elemente des                           
wisenschaftlichen ,denkens1 ,leiden1    
vol4 i41 #ahij2               vol4 ii41 
#ahid4    ,h4 ,j4     ,the ,hibbert     
,journal3 a ,quarterly ,review of       
,religion1               ,theology and  
,philosophy1 ,london and ,new ,york4    
,h4 ,s4     ,lewis ,carroll3 ,the       

,hunting of the ,snark3 an ,agony in    
,eight ,fits1 ,london1 #aiaa4    ,m4    
,the ,monist3 a ,quarterly ,magazine    
,devoted to ,science and                
,philosophy1 ,chicago and ,london4      
,md4       ,mind3 a ,quarterly ,review  
of ,psychology and ,philosophy1         
,london and ,new ,york4    ,pa4 ,ma4    
,alfred ,north ,whitehead and ,bertrand 
,russell3 ,principia                    
,mathematica1 vol4 i41 ,cambridge1      
#aiaj4 ,other volumes               were
 published in #aiab and #aiac4    ,p4   
,e4     ,bertrand ,russell3             
,philosophical ,essays1 ,london and ,new
               ,york1 #aiaj4    ,ph4 ,l4
    ,bertrand ,russell3 ,a ,critical    
,exposition of the ,philosophy          
of ,leibniz1 with an ,appendix of       
,leading ,passages1                     
,cambridge1 #aijj4    ,p4 ,m4           
,bertrand ,russell3 ,the ,principles of 
,mathematics1               vol4 i41    
,cambridge1 #aijc4    ,r4 ,m4 ,m4       
,revue de ,metaphysique et de ,morale1  

,paris4    ,s4 ,b4     ,lewis ,carroll3 
,sylvie and ,bruno1 ,london1 #ahhi4     
,s4 ,l4     ,john ,venn3 ,symbolic      
,logic1 ,london1 #ahha2 #bnd ed41 #ahid4
    ,s4 o4 ,s4  ,william ,stanley       
,jevons3 ,the ,substitution of          
,similars1 the               ,true      
,principle of ,reasoning derived from a 
,modification of                        
,aristotle's ,dictum1 ,london1 #ahfi4   
,t4 ,l4 ,g4  ,lewis ,carroll3 ,through  
the ,looking-,glass1 and what ,alice    
found there1 ,london1 #aiaa4    ,z4 ,s4 
,gottlob ,frege3 ,ueber die ,zahlen des 
,herrn ,h4 ,schubert1                   
,jena1 #ahii4     ,,chapter ,i  ,,the   
,,indefinables ,,of ,,logic   ,the view 
that the fundamental principles of logic
 consist solely of the law of identity  
was held by ,leibniz1#b ,drobisch1      
,uberweg1#c and ,tweedledee4            
,tweedledee1 it may be remembered1#d    
remarked that certain identities are    
logic4 ,now1 there is some doubt as to  
whether he1 like ,jevons1#e understood  

are to mean what mathematicians mean by 
1 or1 like ,schroeder#f and most        
logicians1 to have the same meaning as  
the relation of subsumption4 ,the first 
alternative alone would justify our     
contention2 and we may1 ,i think1       
conclude from an opposition to authority
 that may have been indicated by        
,tweedledee's frequent use of the word  
contrariwise that he did not follow the 
majority of logicians1 but held1 like   
,jevons1#g the mistaken#h view that the 
quantification of the predicate is      
relevant to symbolic logic4  ,it may be 
mentioned1 by the way1 that it is       
probable that ,humpty-,dumpty's is is   
the is of identity4 ,in fact1 it is not 
unlikely that ,humpty-,dumpty was a     
,hegelian2 for1 although his ability for
 clear explanation may seem to militate 
against this1 yet his inability to      
understand mathematics1#i together with 
his synthesis of a cravat and a belt1   
which usually serve different           
purposes1#aj and his proclivity towards 

riddles seem to make out a good case for
 those who hold that he was in fact a   
,hegelian4 ,indeed1 riddles are very    
closely allied to puns1 and it was upon 
a pun1 consisting of the confusion of   
the is of predication with the is of    
identity--so that1 for example1         
,socrates was identified with mortal and
 more generally the particular with the 
universal--that ,hegel's system of      
philosophy was founded4#aa ,but the     
question of ,humpty-,dumpty's           
philosophical opinions must be left for 
final verification to the historians of 
philosophy3 here ,i am only concerned   
with an a priori logical construction of
 what his views might have been if they 
formed a consistent whole4#ab  ,if the  
principle of identity were indeed the   
sole principle of logic1 the principles 
of logic could hardly be said to be1 as 
in fact they are1 a body of propositions
 whose consistency it is impossible to  
prove4#ac ,this characteristic is       
important and one of the marks of the   

greatest possible security4 ,for        
example1 while a great achievement of   
late years has been to prove the        
consistency of the principles of        
arithmetic1 a science which is          
unreservedly accepted except by some    
empiricists1#ad it can be proved        
formally that one foundation of         
arithmetic is shattered4#ae ,it is true 
that1 quite lately1 it has been shown   
that this conclusion may be avoided1    
and1 by a re-moulding of logic1 we can  
draw instead the paradoxical conclusion 
that the opinions held by common-sense  
for so many years are1 in part1         
justified4 ,but it is quite certain     
that1 with the principles of logic1 no  
such proof of consistency1 and no such  
paradoxical result of further           
investigations is to be feared4  ,still1
 this re-moulding has had the result of 
bringing logic into a fuller agreement  
with common-sense than might be         
expected4 ,there were only two          
alternatives3 if we chose principles in 

accordance with common-sense1 we arrived
 at conclusions which shocked           
common-sense2 by starting with          
paradoxical principles1 we arrived at   
ordinary conclusions4 ,like the ,white  
,knight1 we have dyed our whiskers an   
unusual colour and then hidden them4#af 
,the quaint name of ,laws of ,thought1  
which is often applied to the principles
 of ,logic1 has given rise to confusion 
in two ways3 in the first place1 the    
,laws1 unlike other laws1 cannot be     
broken1 even in thought2 and1 in the    
second place1 people think that the     
,laws have something to do with holding 
for the operations of their minds1 just 
as laws of nature hold for events in the
 world around us4#ag ,but that the laws 
are not psychological laws follows from 
the facts that a thing may be true even 
if nobody believes it1 and something    
else may be false if everybody believes 
it4 ,such1 it may be remarked1 is       
usually the case4  ,perhaps the most    
frequent instance of the assumption that

 the laws of logic are mental is the    
treatment of an identity as if its      
validity were an affair of our          
permission4 ,some people suggest to     
others that they should let bygones be  
bygones4 ,another important piece of    
evidence that the truth of propositions 
has nothing to do with mind is given by 
the phrase it is morally certain that   
such-and-such a proposition is true4    
,now1 in the first place1 morality1     
curiously enough1 seems to be closely   
associated with mental acts3 we have    
professorships and lectureships of1 and 
examinations in1 mental and moral       
philosophy4 ,in the second place1 it is 
plain that a morally certain proposition
 is a highly doubtful one4 ,thus it is  
as vain to expect any information about 
our minds from a study of the ,laws of  
,thought as it would be to expect a     
description of a certain social event   
from ,miss ,e4 ,e4 ,c4 ,jones's book ,an
 ,introduction to ,general ,logic4      
,fortunately1 the principles or laws of 

,logic are not a matter of philosophical
 discussion4 ,idealists like ,tweedledum
 and ,tweedledee1 and even practical    
idealists like the ,white ,knight1      
explicitly accept laws like the law of  
identity and the excluded middle4#ah ,in
 fact1 throughout all logic and         
mathematics1 the existence of the human 
or any other mind is totally irrelevant2
 mental processes are studied by means  
of logic1 but the subject-matter of     
logic does not presuppose mental        
processes1 and would be equally true if 
there were no mental processes4 ,it is  
true that1 in that case1 we should not  
know logic2 but our knowledge must not  
be confounded with the truths which we  
know4#ai ,an apple is not confused with 
the eating of it except by savages1     
idealists1 and people who are too hungry
 to think4  ,,footnotes3  #b ,russell1  
,ph4 ,l41 pp4 #ag1 #ai1 #bjg-h4  #c     
,schroeder1 ,a4 d4 ,l41 i4 p4 #d4  #d   
,see ,appendix ,a4 ,this ,appendix also 
illustrates the importance attached to  

the ,principle of ,identity by the      
,professor and ,bruno4  #e ,s4 o4 ,s41  
pp4 #i-ae4  #f ,a4 d4 ,l41 i4 p4 #acb4  
#g ,cf41 besides the reference in the   
last note but one1 ,e4 ,l4 ,l41 pp4     
#ahc1 #aia4 ,contrariwise1 it may be    
remarked1 is not a term used in         
traditional logic4  #h ,s4 ,l41 #ahha1  
pp4 #agc-e1 #cbd-e2 #ahid1 pp4 #aid-f4  
#i ,cf4 ,appendix ,c1 and ,william      
,robertson ,smith1 ,hegel and the       
,metaphysics of the ,fluxional          
,calculus1 ,trans4 ,roy4 ,soc41 ,edinb41
 vol4 xxv41 #ahfi1 pp4 #dia-eaa4  #aj   
,see ,appendix ,b4  #aa ,this is a      
remarkable anticipation of the note on  
pp4 #ci-dj of ,mr4 ,russell's book1     
published about three years after the   
death of ,mr4 ,r99ss99ll1 and entitled  
,our ,knowledge of the ,external ,world 
as a ,field for ,scientific ,method in  
,philosophy1 ,chicago and ,london1      
#aiad4--;,,ed4  #ab ,cf4 ,ph4 ,l41 pp4  
v4-vi4 #c4  #ac ,cf4 ,pieri1 ,r4 ,m4    
,m41 ,march #aijf1 p4 #aii4  #ad ,as a  

type of these1 ,humpty-,dumpty1 with his
 inability to admit anything not        
empirically given and his lack of       
comprehension of pure mathematics1 may  
be taken 7see ,appendix ,c74 ,in his    
7correct7 thesis that definitions are   
nominal1 too1 ,humpty-,dumpty reminds   
one of ,j4 ,s4 ,mill 7see ,appendix ,d74
  #ae ,see ,frege1 ,gg41 ii4 p4 #bec4   
#af ,see ,appendix ,e4  #ag ,see ,frege1
 ,gg41 i4 p4 #ae4  #ah ,see the above   
references and also ,appendix ,f4  #ai  
,cf4 ,b4 ,russell1 ,h4 ,j41 ,july #aijd1
 p4 #hab4     ,,chapter ,,ii            
,,objective ,,validity ,,of ,,the ,,laws
 ,,of ,,thought   ,i once inquired of a 
maid-servant whether her mistress was at
 home4 ,she replied1 in a doubtful      
fashion1 that she thought that her      
mistress was in unless she was out4 ,i  
concluded that the maid was uncertain as
 to the objective validity of the law of
 excluded middle1 and remarked that to  
her mistress4 ,but since ,i used the    
phrase laws of thought1 the mistress    

perhaps supposed that a law of thought  
has something to do with thinking and   
seemed to imagine that ,i wished to     
impute to the maid some moral defect of 
an unimportant nature4 ,thus she        
remonstrated with me in an amused way1  
since she probably imagined that ,i     
meant to find fault with the maid's     
capacity for thinking4     ,,chapter    
,,iii  ,,identity   ,in the first       
chapter we have noticed the opinion that
 identities are fundamental to all      
logic4 ,we will now consider some other 
views of the value of identities4       
,identities are frequently used in      
common life by people who seem to       
imagine that they can draw important    
conclusions respecting conduct or       
matters of fact from them4 ,i have heard
 of a man who gained the double         
reputation of being a philosopher and a 
fatalist by the repeated enunciation of 
the identity ,whatever will be1 will be2
 and the ,italian equivalent of this    
makes up an appreciable part of one of  

,mr4 ,robert ,hichens' novels4 ,further1
 the identity ,life is ,life has not    
only been often accepted as an          
explanation for a particular way of     
living but has even been considered by  
an authoress who calls herself ,zack to 
be an appropriate title for a novel2    
while ,business is ,business is         
frequently thought to provide an excuse 
for dishonesty in trading1 for which    
purpose it is plainly inadequate4       
,another example is given by a poem of  
,mr4 ,kipling1 where he seems to assert 
that ,east is ,east and ,west is ,west  
imply that never the twain shall meet4  
,the conclusion1 now1 is false2 for1    
since the world is round--as geography  
books still maintain by arguments which 
strike every intelligent child as       
invalid#bj--what is called the ,west    
does1 in fact1 merge into the ,east4    
,even if we are to take the statement   
metaphorically1 it is still untrue1 as  
the ,japanese nation has shown4  ,the   
law-courts are often rightly blamed for 

being strenuous opponents of the spread 
of modern logic3 the frequent misuse of 
and1 or1 the1 and provided that in them 
is notorious4 ,but the fault seems      
partly to lie in the uncomplicated      
nature of the logical problems which are
 dealt with in them4 ,thus it is no     
uncommon thing for somebody to appear   
there who is unable to establish his own
 identity1 or for ,a to assert that ,b  
was not himself when he made a will     
leaving his money to ,c4  ,the chief use
 of identities is in implication4       
,since1 in logic1 we so understand      
implication#ba that any true proposition
 implies and is implied by any other    
true proposition2 if one is convinced of
 the truth of the proposition ,q1 it is 
advisable to choose one or more         
identities ,p1 whose truth is undoubted1
 and say that ,p implies ,q4 ,thus1 ,mr4
 ,austen ,chamberlain1 according to ,the
 ,times of ,march #bg1 #aiji1 professed 
to deduce the conclusion that it is not 
right that women should have votes from 

the premisses that man is man and woman 
is woman4 ,this method requires that one
 should have made up one's mind about   
the conclusion before discovering the   
premisses--by what1 no doubt1 ,jevons   
would call an inverse or inductive      
method4 ,thus the method is of use only 
in speeches and in giving good advice4  
,mr4 ,austen ,chamberlain afterwards    
rather destroyed one's belief in the    
truth of his premisses by putting limits
 to the validity of the principle of    
identity4 ,in the course of the ,debate 
on the ,budget of #aiji1 he maintained1 
against ,mr4 ,lloyd ,george1 that a joke
 was a joke except when it was an       
untruth3 ,mr4 ,lloyd ,george1           
apparently1 being of the plausible      
opinion that a joke is a joke under all 
circumstances4  ,,footnotes3  #bj ,the  
argument about the hull of a ship       
disappearing first is not convincing1   
since it would equally well prove that  
the surface of the earth was1 for       
example1 corrugated on a large scale4   

,if the common-sense of the reader were 
supposed to dismiss the possibility of  
water clinging to such corrugations1 it 
might equally be supposed to dismiss the
 possibility of water clinging to a     
spherical earth4 ,traditional geography 
books1 no doubt1 gave rise to the       
opinions held by ,lady ,blount and the  
,zetetic ,society4  #ba ,the subject of 
,implication will be further considered 
in ,chapter ,,xix4     ,,chapter ,,iv   
,,identity ,,of ,,classes   ,i once     
heard of a meritorious lady who was     
extremely conventional2 on the slender  
grounds of carefully acquired habits of 
preferring the word woman to the word   
lady and of going to the post-office    
without a hat1 imagined that she was    
unconventional and altogether a         
remarkable person2 and who once remarked
 with great satisfaction that she was a 
very queer person1 and that nothing     
shocked her except1 of course1 bad form4
  ,thus1 she asserted that all the      
things which shocked her were actions in

 bad form2 and she would undoubtedly    
agree1 though she did not actually state
 it1 that all the things which were done
 in bad form would shock her4           
,consequently she asserted that the     
class of things which shocked her was   
the class of actions in bad form4       
,consequently the statement of this lady
 that some or all of the actions done in
 bad form shocked her is an identical   
proposition of the form nothing shocks  
me1 except1 of course1 the things which 
do1 in fact1 shock me2 and this         
statement the lady certainly did not    
intend to make4  ,this excellent lady1  
had she but known it1 was logically     
justified in making any statement       
whatever about her unconventionality4   
,for the class of her unconventional    
actions was the null class4 ,thus she   
might logically have made inconsistent  
statements about this class of actions4 
,as a matter of fact she did make       
inconsistent statements1 but            
unfortunately she justified them by     

stating that1 ,it is the privilege of   
woman to be inconsistent4 ,she was one  
of those persons who say things like    
that4     ,,chapter ,v  ,,ethical       
,,applications ,,of ,,the ,,law ,,of    
,,identity   ,it may be remembered that 
,mr4 ,podsnap remarked1 with sadness    
tempered by satisfaction1 that he       
regretted to say that ,foreign nations  
do as they do do4 ,besides aiding the   
comforting expression of moral          
disapproval1 the law of identity has yet
 another useful purpose in practical    
ethics3 ,it serves the welcome purpose  
of providing an excuse for infractions  
of the moral law4 ,there was once a man 
who treated his wife badly1 was         
unfaithful to her1 was dishonest in     
business1 and was not particular in his 
use of language2 and yet his life on    
earth was described in the lines3       
,this man maintained a wife's a wife1   
,men are as they are made1   ,business  
is business1 life is life2   ,and called
 a spade a spade4  ,one of the objects  

of ,dr4 ,g4 ,e4 ,moore's ,principia     
,ethica#bb was to argue that the word   
good means simply good1 and not pleasant
 or anything else4 ,appropriately       
enough1 this book bore on its title-page
 the quotation from the preface to the  
,sermons1 published in #agbf1 of ,bishop
 ,joseph ,butler1 the author of the     
,analogy3 ,everything is what it is and 
not another thing4  ,but another famous 
,butler--,samuel ,butler1 the author of 
,hudibras--went farther than this1 and  
maintained that identities were the     
highest attainment of metaphysics       
itself4 ,at the beginning of the first  
,canto of ,hudibras1 in the description 
of ,hudibras himself1 ,butler wrote3    
,he knew what's what1 and that's as high
   ,as metaphysic wit can fly4  ,i once 
conducted what ,i imagined to be an     
aesthetic investigation for the purpose 
of discovery1 by the continual use of   
the word ,why8#bc the grounds upon which
 certain people choose to put milk into 
a tea-cup before the tea4 ,i was        

surprised to discover that it was an    
ethical1 and not an aesthetic problem2  
for ,i soon elicited the fact that it   
was done because it was right4 ,a       
continuance of my patient questioning   
elicited further evidence of the        
fundamental character of the principle  
of identity in ethics2 for it was right1
 ,i learned1 because right is right4    
,it appears that some people            
unconsciously think that the principle  
of identity is the foundation1 in       
certain religions1 of the reasons which 
can be alleged for moral conduct1 and   
are surprised when this fact is pointed 
out to them4 ,the late ,sir ,leslie     
,stephen1 when travelling by railway1   
fell into conversation with an officer  
of the ,salvation ,army1 who tried hard 
to convert him4 ,failing in this        
laudable endeavour1 the ,salvationist at
 last remarked3 ,but if you aren't      
saved1 you can't go to heaven6 ,that1 my
 friend1 replied ,stephen1 is an        
identical proposition4  ,,footnotes3    

#bb ,cambridge1 #aijc4  #bc ,cf4 ,p4    
,e41 p4 #b4     ,,chapter ,,vi  ,,the   
,,law ,,of ,,contradiction ,,in ,,modern
 ,,logic   ,considering the important   
place assigned by philosophers and      
logicians to the law of contradiction1  
the remark will naturally be resented by
 many of the older schools of           
philosophy1 and especially by ,kantians1
 that in spite of its fame we have found
 few occasions for its use4#bd ,also in 
modern times1 ,benedetto ,croce1 an     
opponent of both traditional logic and  
mathematical logic1 began the preface of
 the book of #aijh on ,logic#be by      
saying that that volume is and is not a 
certain memoir of his which had been    
published in #aije4  ,,footnotes3  #bd  
,pa4 ,ma41 p4 #aaf4  #be ,english       
translation of the third ,italian       
edition by ,douglas ,ainslie1 under the 
title3 ,logic as the ,science of the    
,pure ,concept1 ,london #aiag4--;,,ed4  
,,chapter ,,vii  ,,symbolism ,,and      
,,meaning   ,when people write down any 

statement such as ,the curfew tolls the 
knell of parting day1#bf which we will  
call ,c for shortness1 what they mean is
 not ,c but the meaning of ,c2 and not  
the meaning of ',c' but the meaning of  
the meaning of ',c'4 ,and so on1 ad     
infinitum4 ,thus1 in writing or in      
speech1 we always fail to state the     
meaning of any proposition whatever4    
,sometimes1 indeed1 we succeed in       
conveying it2 but there is danger in too
 great a disregard of statement and     
preoccupation with conveyance of        
meaning4 ,thus many mathematicians have 
been so anxious to convey to us a       
perfectly distinct and unmetaphysical   
concept of number that they have        
stripped away from it everything that   
they considered unessential 7like its   
logical nature7 and have finally        
delivered it to us as a mere sign4 ,by  
the labours of ,helmholtz1 ,kronecker1  
,heine1 ,stolz1 ,thomae1 ,pringsheim1   
and ,schubert1 many people were         
persuaded that1 when they said '#b' is a

 number they were speaking the truth1   
and hold that ,paris is a town          
containing the letter ,p4 ,when ,frege  
pointed out#bg this difficulty he was   
almost universally denounced in ,germany
 as spitzfindig4 ,in fact1 ,germans seem
 to have been influenced perhaps by that
 great contemner of ,spitzfindigkeit1   
,kant1 to reject the ,white ,knight's#bh
 distinctions between words and their   
denotations and to regard subtlety with 
disfavour to such a degree that their   
only mathematical logician except       
,frege1 namely ,schroeder--the least    
subtle of mortals1 by the way--seems to 
have been filled with such fear of being
 thought subtle1 that he made his books 
so prolix that nobody has read them4    
,another term which1 as we shall see    
when discussing the paradoxes of logic1 
mathematicians are accustomed to apply  
to thought which is more exact than any 
to which they are accustomed is         
scholastic4#bi ,by this1 ,i suppose1    
they mean that the pursuits of certain  

acute people of the ,middle ,ages are   
unimportant in contrast with the great  
achievements of modern thought1 as      
exemplified by a method of making       
plausible guesses known as induction1#cj
 the bicycle1 and the gramophone--all of
 them instruments of doubtful merit4    
,,footnotes3  #bf ,cf4 ,md1 ,n4 ,s41    
vol4 xiv41 ,october #aije1 p4 #dhf4  #bg
 ,in ,z4 ,s41 for example4  #bh ,see    
,appendix ,g4  #bi ,cf4 ,chapter        
,,xxxvii below4  #cj ,cf4 ,p4 ,m41 p4   
#aa1 note4     ,,chapter ,,viii         
,,nominalism   ,de ,morgan#ca said that1
 if all mankind had spoken one language1
 we cannot doubt that there would have  
been a powerful1 perhaps universal1     
school of philosophers who would have   
believed in the inherent connexion      
between names and things2 who would have
 taken the sound man to be the mode of  
agitating the air which is essentially  
communicative of the ideas of reason1   
cookery1 bipedality1 etc4444 ',the      
,french1' said the sailor1 'call a      

cabbage a shoe2 the fools6 ,why can't   
they call it a cabbage1 when they must  
know it is one8'  ,one of the chief     
differences between logicians and men of
 letters is that the latter mean many   
different things by one word1 whereas   
the former do not--at least nowadays4   
,most mathematicians belong to the class
 of men of letters4  ,i once had a      
manservant who told me on a certain     
occasion that he never thought a word   
about it4 ,i was doubtful whether to    
class him with such eminent             
mathematicians as are mentioned in the  
last chapter1 or as a supporter of ,max 
,mueller's theory of the identity of    
thought and language4 ,however1 since   
the man was very untruthful1 and he told
 me that he meant what he said and said 
what he meant1#cb the conclusion is     
probably correct that he really believed
 that the meanings of his words were not
 the words themselves4 ,thus ,i think it
 most probable that my manservant had   
been a mathematician but had escaped by 

the aid of logic4  ,as regards his      
remark that he meant what he said and   
said what he meant1 he plainly wished to
 pride himself on certain virtues which 
he did not possess1 and was not         
indifferent to applause1 which1 however1
 was never evoked4 ,the virtues1 if so  
they be1 and the applause were withheld 
for other reasons than that the above   
statements are either nonsensical or    
false4 ,suppose that ,i say what ,i mean
 expresses a truth4 ,what ,i say 7or    
write7 is always a symbol--words 7or    
marks72 and what ,i mean by the symbol  
is the meaning of the symbol and not the
 symbol itself4 ,so the remark cannot   
express a truth1 any more than the name 
,wellington won the battle of ,waterloo4
  ,,footnotes3  #ca ,f4 ,l41 pp4 #bdf-g4
  #cb ,the ,hatter 7see ,appendix ,h7   
pointed out that there is a difference  
between these two assertions4 ,thus1 he 
clearly showed that he was a nominalist1
 and philosophically opposed to the     
,march ,hare who had recommended ,alice 

to say what she meant4     ,,chapter    
,,ix  ,,ambiguity ,,and ,,symbolic      
,,logic   ,the universal use of some    
system of ,symbolic ,logic would not    
only enable everybody easily to deal    
with exceedingly complicated arguments1 
but would prevent ambiguous arguments4  
,in denying the indispensability of     
,symbolic ,logic in the former state of 
things1 ,keynes#cc is probably alone1   
against the need strongly felt by ,alice
 when speaking to the ,duchess1#cd and  
most modern logicians4 ,it may be       
noticed that the ,duchess is more       
consistent than ,keynes1 for ,keynes    
really uses the signs for logical       
multiplication and addition of ,boole   
and ,venn under the different shapes of 
the words and and or4  ,as regards      
ambiguity1 a translation of ,hymns      
,ancient and ,modern into1 say1         
,peanesque1 would prevent the puzzle of 
childhood as to whether the his in      
,and ,satan trembles when he sees   ,the
 weakest saint upon his knees  refers to

 the saint's knees or ,satan's4         
,,footnotes3  #cc ,in his ,fm4 ,l4  #cd 
,see ,appendix ,i4     ,,chapter ,x     
,,logical ,,addition ,,and ,,the        
,,utility ,,of ,,symbolism   ,frequently
 ordinary language contains subtle      
psychological implications which cannot 
be translated into symbolic logic except
 at great length4 ,thus if a man 7say   
,mr4 ,jones7 wishes to speak            
collectively of himself and his wife1   
the order of mentioning the terms in the
 class considered and the names applied 
to these terms are1 logically speaking1 
irrelevant4 ,and yet more or less       
definite information is given about ,mr4
 ,jones according as he talks to his    
friends of3       7#a7 ,mrs4 ,jones and 
,i1      7#b7 ,i 7or me7 and my wife 7or
 missus71      7#c7 ,my wife and ,i1    
or 7#d7 ,i 7or me7 and ,mrs4 ,jones4    
,in case 7#a7 one is probably correct in
 placing ,mr4 ,jones among the clergy or
 the small professional men who make up 
the bulk of the middle-class2 in case   

7#b7 one would conclude that ,mr4 ,jones
 belonged to the lower middle-class2 the
 form 7#c7 would be used by ,mr4 ,jones 
if he were a member of the upper1 upper 
middle1 or lower class2 while form 7#d7 
is only used by retired shopkeepers of  
the lower middle-class1 of which a male 
member usually combines belief in the   
supremacy of man with belief in the     
dignity of his wife as well as himself4 
,a further complication is introduced if
 a wife is referred to as the wife4#ce  
,cases 7#b7 and 7#c7 then each give rise
 to one more case4 ,cases 7#a7 and 7#d7 
do not1 since nobody has hitherto       
referred to his wife as the ,mrs4       
,jones--at least without a qualifying   
adjective before the ,mrs4  ,on the     
other hand1 certain descriptive phrases 
and certain propositions can be         
expressed more shortly and more         
accurately by means of symbolic logic4  
,let us consider the proposition ,no man
 marries his deceased wife's sister4 ,if
 we assume1 as a first approximation1   

that all marriages are fertile and that 
all children are legitimate1 then1 with 
only four primitive ideas3 the relation 
of parent to child 7,p7 and the three   
classes of males1 females1 and dead     
people1 we can define wife 7a female who
 has the relation formed by taking the  
relative product of ,p and v,p#cf to a  
male71 sister1 deceased wife1 and       
deceased wife's sister in terms of these
 ideas and of the fundamental notions of
 logic4 ,then the proposition ,no man   
marries his deceased wife's sister can  
be expressed unambiguously by about     
twenty-nine simple signs on paper1      
whereas1 in words1 the unasserted       
statement consists of no less than      
thirty-four letters4 ,although1 legally 
speaking1 we should have to adopt       
somewhat different definitions and      
possibly increase the complications of  
our proposition1 it must be remembered  
that1 on the other hand1 we always      
reduce the number of symbols in any     
proposition by increasing the number of 

definitions in the preliminaries to it4 
,but the utility of symbolic logic      
should not be estimated by the brevity  
with which propositions may sometimes be
 expressed by its means4 ,logical       
simplicity1 in fact1 can very often only
 be obtained by apparently complicated  
statements4 ,for example1 the logical   
interpretation of ,the father of        
,charles ,,ii was executed is1 ,it is   
not always false of x that x begat      
,charles ,,ii1 and that x was executed  
and that 'if y begat ,charles ,,ii1 y is
 identical with x' is always true of    
y4#cg ,from the point of view of logic1 
we may say that the apparently simple is
 most often very complicated1 and1 even 
if it is not so1 symbolism will make it 
seem so1#ch and thus draw attention to  
what might otherwise easily be          
overlooked4  ,,footnotes3  #ce ,cf4     
,chapter ,,xxiv below4  #cf ,c4 ,s4     
,peirce's notation for the relation     
converse of ,p4  #cg ,russell1 ,md41 ,n4
 ,s41 vol4 xiv41 ,october #aije1 p4     

#dhb4  #ch ,russell1 ,international     
,monthly1 vol4 iv41 #aija1 pp4 #he-f2   
cf4 ,m41 vol4 xxii41 #aiab1 p4 #aec4    
,this essay is reprinted in ,mysticism  
and ,logic1 ,london and ,new ,york1     
#aiah1 pp4 #gd-if4--;,,ed4     ,,chapter
 ,,xi  ,,criticism   ,those people who  
think that it is more godlike to seem to
 turn water into wine than to seem to   
turn wine into water surprise me4 ,i    
cannot imagine an intolerable critic4   
,it seems to me that1 if ,a resents ,b's
 criticism in trying to put his 7,a's7  
discovery in the right or wrong place1  
,a acts as if he thought he had some    
private property in truth4 ,the ,white  
,queen seems to have shared the popular 
misconception as to the nature of       
criticism4#ci  ,,footnotes3  #ci ,see   
,appendix ,j4     ,,chapter ,,xii       
,,historical ,,criticism   ,from a      
problem in ,diophantus's ,arithmetic    
about the price of some wine it would   
seem that the wine was of poor quality1 
and ,paul ,tannery has suggested that   

the prices mentioned for such a wine are
 higher than were usual until after the 
end of the second century4 ,he therefore
 rejected the view which was formerly   
held that ,diophantus lived in that     
century4#dj  ,the same method applied to
 a problem given by the ancient ,hindu  
algebraist ,brahmagupta1 who lived in   
the seventh century after ,christ1 might
 result in placing ,brahmagupta in      
prehistoric times4 ,this is the         
problem3#da ,two apes lived at the top  
of a cliff of height h1 whose base was  
distant mh from a neighbouring village4 
,one descended the cliff and walked to  
the village1 the other flew up a height 
x and then flew in a straight line to   
the village4 ,the distance traversed by 
each was the same4 ,find x4             
,,footnotes3  #dj ,w4 ,w4 ,rouse ,ball1 
,a ,short ,account of the ,history of   
,mathematics1 #dth edition1 ,london1    
#aijh1 p4 #aji4  #da ,ibid41 pp4 #adh-i4
     ,,chapter ,,xiii  ,,is ,,the ,,mind
 ,,in ,,the ,,head8   ,the contrary     

opinion has been maintained by idealists
 and a certain election agent with whom 
,i once had to deal and who remarked    
that something slipped his mind and then
 went out of his head altogether4 ,at   
some period1 then1 a remembrance was in 
his head and out of his mind2 his mind  
was not1 then1 wholly within his head4  
,also1 one is sometimes assured that    
with certain people out of sight is out 
of mind4 ,what is in their minds is     
therefore in sight1 and cannot therefore
 be inside their heads4     ,,chapter   
,,xiv  ,,the ,,pragmatist ,,theory ,,of 
,,truth   ,the pragmatist theory that   
truth is a belief which works well      
sometimes conflicts with common-sense   
and not with logic4 ,it is commonly     
supposed that it is always better to be 
sometimes right than to be never right4 
,but this is by no means true4 ,for     
example1 consider the case of a watch   
which has stopped2 it is exactly right  
twice every day4 ,a watch1 on the other 
hand1 which is always five minutes slow 

is never exactly right4 ,and yet there  
can be no question but that a belief in 
the accuracy of the watch which was     
never right would1 on the whole1 produce
 better results than such a belief in   
the one which had altogether stopped4   
,the pragmatist would1 then1 conclude   
that the watch which was always         
inaccurate gave truer results than the  
one which was sometimes accurate4 ,in   
this conclusion the pragmatist would    
seem to be correct1 and this is an      
instance of how the false premisses of  
pragmatism may give rise to true        
conclusions4  ,from the text written    
above the church clock in a certain     
,english village1 ,be ye ready1 for ye  
know not the time1 it would be concluded
 that the clock never stopped for a     
period as long as twelve hours4 ,for the
 text is rather a vague symbolical      
expression of a propositional function  
which is asserted to be true at all     
instants4 ,the proposition that a       
presumably not illiterate and credulous 

observer of the clock at any definite   
instant does not know the time implies1 
then1 that the clock is always wrong4   
,now1 if the clock stopped for twelve   
hours1 it would be absolutely right at  
least once4 ,it must be right twice if  
it were right at the first instant it   
stopped or the last instant at which it 
went2#db but the second possibility is  
excluded by hypothesis1 and the         
occurrence of the first possibility--or 
of the analogous possibility of the     
stopped clock being right three times in
 twenty-four hours--does not affect the 
present question4 ,hence the clock can  
never stop for twelve hours4  ,the      
pragmatist's criterion of truth appears 
to be far more difficult to apply than  
the ,bellman's1#dc that what he said    
three times is true1 and to give results
 just as insecure4  ,,footnotes3  #db   
,both cases cannot occur2 the question  
is similar to that arising in the       
discussion of the mortality of ,socrates
 7see ,chapter ,,xxii74  #dc ,see       

,appendix ,k4     ,,chapter ,,xv        
,,assertion   ,the subject of the       
present chapter must not be confused    
with the assertion of ordinary life4    
,commonly1 an unasserted proposition is 
synonymous with a probably false        
statement1 while an asserted proposition
 is synonymous with one that is         
certainly false4 ,but in logic we apply 
assertion also to true propositions1    
and1 as ,lewis ,carroll showed in his   
version of ,what the ,tortoise said to  
,achilles1#dd usually pass over         
unconsciously an infinite series of     
implications in so doing4 ,if p and q   
are propositions1 p is true1 and p      
implies q1 then1 at first sight1 one    
would think that one might assert q4    
,but1 from 7,a7 p is true1 and 7,b7 p   
implies q1 we must1 in order to deduce  
7,z7 q is true1 accept the hypothetical3
 7,c7 ,if ,a and ,b are true1 ,z must be
 true4 ,and then1 in order to deduce ,z 
from ,a1 ,b1 and ,c1 we must accept     
another hypothetical3 7,d7 ,if ,a1 ,b1  

and ,c are true1 ,z must be true2 and so
 on ad infinitum4 ,thus1 in deducing ,z1
 we pass over an infinite series of     
hypotheticals which increase in         
complexity4 ,thus we need a new         
principle to be able to assert q4       
,frege was the first logician sharply to
 distinguish between an asserted        
proposition1 like ,a is greater than ,b1
 and one which is merely considered1    
like ,a's being greater than ,b1        
although an analogous distinction had   
been made in our common discourse on    
certain psychological grounds1 for long 
previously4 ,in fact1 soon after the    
invention of speech1 the necessity of   
distinguishing between a considered     
proposition and an asserted one became  
evident1 on account of the state of     
things referred to at the beginning of  
this chapter4  ,,footnotes3  #dd ,md4   
,n4 ,s41 vol4 iv41 #ahie1 pp4 #bgh-hj4  
,cf4 ,russell1 ,p4 ,m41 p4 #ce4         
,,chapter ,,xvi  ,,the ,,commutative    
,,law   ,often the meaning of a sentence

 tacitly implies that the commutative   
law does not hold4 ,we are all familiar 
with the passage in which ,macaulay     
pointed out that1 by using the          
commutative law because of exigencies of
 metre1 ,robert ,montgomery             
unintentionally made ,creation tremble  
at the ,atheist's nod instead of the    
,almighty's4 ,this use of the           
commutative law by writers of verse     
renders it doubtful whether1 in the     
hymn-line3    ,the humble poor believe1 
we are to understand a statement about  
the humble poor1 or a doubtful maxim as 
to the attitude of our minds to         
statements made by the humble poor4     
,the non-commutativity of ,english      
titles offers difficulties to some      
novelists and ,americans who refer to   
,mary ,lady ,so-and-,so as ,lady ,mary  
,so-and-,so1 and vice versa4            
,,chapter ,,xvii  ,,universal ,,and     
,,particular ,,propositions   ,people   
who are cynical as to the morality of   
the ,english are often unpleasantly     

surprised to learn that ,all trespassers
 will be prosecuted does not necessarily
 imply that some trespassers will be    
prosecuted4 ,the view that universal    
propositions are non-existential is now 
generally held3 ,bradley and ,venn seem 
to have been the first to hold this1    
while older logicians1 such as ,de      
,morgan1#de considered universal        
propositions to be existential1 like    
particular ones4  ,if the ,gnat#df had  
been content to affirm his proposition  
about the means of subsistence of       
,bread-and-,butter flies1 in consequence
 of their lack of which such flies      
always die1 without pointing out such an
 insect and thereby proving that the    
class of them is not null1 ,alice's     
doubt as to the existence of the class  
in question1 even if it were proved to  
be well founded1 would not have affected
 the validity of the proposition4  ,this
 brings us to a great convenience in    
treating universal propositions as      
non-existential3 we can maintain that   

all x's are y's at the same time as that
 no x's are y's1 if only x is the       
null-class4 ,thus1 when ,mr4            
,mac,coll#dg objected to other symbolic 
logicians that their premisses imply    
that all ,centaurs are flower-pots1 they
 could reply that their premisses also  
imply the more usual view that ,centaurs
 are not flower-pots4  ,,footnotes3  #de
 ,cf41 e4g41 ,f4 ,l41 p4 #d4  #df ,see  
,appendix ,l4  #dg ,cf41 e4g41 ,md41 ,n4
 ,s41 vol4 xiv41 ,july1 #aije1 pp4      
#cii-djj4     ,,chapter ,,xviii         
,,denial ,,of ,,generality ,,and        
,,generality ,,of ,,denial   ,the       
conclusion of a certain song#dh about a 
young man who poisoned his sweetheart   
with sheep's-head broth1 and was        
frightened to death by a voice          
exclaiming3    ,where's that young maid 
,what you did poison with my head8  at  
his bedside1 gives rise to difficulties 
which are readily solved by a symbolism 
that brings into relief the principle   
that the denial of a universal and      

non-existential proposition is a        
particular and existential one4 ,the    
conclusion of the song is3    ,now all  
young men1 both high and low1   ,take   
warning by this dismal go6   ,for if    
he'd never done nobody no wrong1   ,he  
might have been here to have heard this 
song4  ,it is an obvious error1 say     
,whitehead and ,russell1#di though one  
easy to commit1 to assume that the      
cases3 7#a7 all the propositions of a   
certain class are true2 and 7#b7 no     
proposition of the class is true2 are   
each other's contradictories4 ,however1 
in the modification#ej of ,frege's      
symbolism which was used by ,russell    
7#a7 is 7x74 x1   and 7#b7 is 7x74 not  
x2  while the contradictory of 7#a7 is3 
not 7x74 x4  ,the last line but one of  
the above verse may1 then1 be written3  
7t74 not 7x74 not not ,greek3 ph 7x1 t71
  where ,greek3 ph 7x1 t7 denotes the   
unasserted propositional function the   
doing wrong to the person x at the      
instant t4 ,by means of the principle of

 double negation we can at once simplify
 the above expression into3    7t74 not 
7x74 ,greek3 ph 7x1 t72  which can be   
thus read3 ,if at every instant of his  
life there was at least one person x to 
whom he did no wrong 7at that instant74 
,it is difficult to imagine any one so  
sunk in iniquity that he would not      
satisfy this hypothesis4 ,we are forced1
 then1 unless our imagination for evil  
is to be distrusted1 to conclude that   
any one might have been there to have   
heard that song4 ,now this conclusion is
 plainly false1 possibly on physical    
grounds1 and certainly on aesthetic     
grounds4 ,it may be added1 by the way1  
that it is quite possible that ,de      
,morgan was mistaken in his             
interpretation of the above proposition 
owing to the fact that he was           
unacquainted with ,frege's work4 ,in    
fact1 if he had not noticed the fact    
that any two of the not's cannot be     
cancelled against one another he would  
have concluded that the interpretation  

was3 ,if he had never done any wrong to 
anybody4  ,according as the symbol for  
not comes before the 7x7 or between the 
7x7 and the ,greek3 ph1 we have an      
expression of what ,frege called        
respectively the denial of generality1  
and the generality of denial4 ,the      
denial of the generality of a denial is 
the form of all existential             
propositions1 while the assertion of or 
denial of generality is the general form
 of all non-existential or universal    
propositions4  ,,footnotes3  #dh ,to    
which ,de ,morgan drew attention in a   
letter2 see 7,mrs47 ,s4 ,e4 ,de ,morgan1
 ,memoir of ,augustus ,de ,morgan1      
,london1 #ahhb1 p4 #cbd4  #di ,pa4 ,ma41
 p4 #af4  #ej ,however1 here1 for the   
printer's convenience1 we depart from   
,mr4 ,russell's usage so far as to write
 not for a curly minus sign4            
,,chapter ,,xix  ,,implication   ,a good
 illustration of the fact that what is  
called implication in logic is such that
 a false proposition implies any other  

proposition1 true or false1 is given by 
,lewis ,carroll's puzzle of the three   
barbers4#ea  ,allen1 ,brown1 and ,carr  
keep a barber's shop together2 so that  
one of them must be in during working   
hours4 ,allen has lately had an illness 
of such a nature that1 if ,allen is out1
 ,brown must be accompanying him4       
,further1 if ,carr is out1 then1 if     
,allen is out1 ,brown must be in for    
obvious business reasons4 ,the problem  
is1 may ,carr ever go out8  ,putting p  
for ,carr is out1 q for ,allen is out   
and r for ,brown is out1 we have3       
7#a7 q implies r1   7#b7 p implies that 
q implies not-r4  ,lewis ,carroll       
supposed that q implies r and q implies 
not-r are inconsistent1 and hence that p
 must be false4 ,but these propositions 
are not inconsistent1 and are1 in fact1 
both true if q is false4 ,the           
contradictory of q implies r is q does  
not imply r which is not a consequence  
of q implies not-r4 ,it seems to be true
 theoretically that1 if ,mr4 ,x is a    

,christian1 he is not an ,atheist1 but  
we cannot conclude from this alone that 
his being a ,christian does not imply   
that he is an ,atheist1 unless we assume
 that the class of ,christians is not   
null4 ,thus1 if p is true1 q is false2  
or1 if ,carr is out1 ,allen is in4 ,the 
odd part of this conclusion is that it  
is the one which common-sense would have
 drawn in that particular case4  ,a     
distinguished philosopher 7,m7 once     
thought that the logical use of the word
 implication--any false proposition     
being said to imply any proposition true
 or false--is absurd1 on the grounds    
that it is ridiculous to suppose that   
the proposition #b and #b make #e       
implies the proposition ,m is the ,pope4
 ,this is a most unfortunate instance1  
because it so happens that the false    
proposition that #b and #b make #e can  
rigorously be proved to imply that ,m1  
or anybody else other than the ,pope1 is
 the ,pope4 ,for if #b and #b make #e1  
since they also make #d1 we would       

conclude that #e is equal to #d4        
,consequently1 subtracting #c from both 
sides1 we conclude that #b would be     
equal to #a4 ,but if this were true1    
since ,m and the ,pope are two1 they    
would be one1 and obviously then ,m     
would be the ,pope4  ,the principle that
 the false implies the true has very    
important applications in political     
arguments4 ,in fact1 it is hard to find 
a single principle of politics of which 
false propositions are not the main     
support4  ,if p and q are two           
propositions1 and p implies q2 then1 if1
 and only if1 q and p are both false or 
both true1 we also have3 q implies p4   
,the most important applications of this
 invertibility were made by the late    
,samuel ,butler#eb and ,mr4 ,g4 ,b4     
,shaw4 ,a political application may be  
made as follows3 ,in a country where    
only those with middling-sized incomes  
are taxed1 conservative and bourgeois   
politicians would still maintain that   
the proposition the rich are taxed      

implies the proposition the poor are    
taxed1 and this implication1 which is   
true because both premiss and conclusion
 are false1 would be quite unnecessarily
 supported by many false practical      
arguments4 ,it is equally true that the 
poor are taxed implies that the rich are
 taxed4 ,and this can be proved1 in     
certain cases1 on other grounds4 ,for   
the taxation of the poor would imply1   
ultimately1 that the poor could not     
afford to pay a little more for the     
necessities of life than1 in strict     
justice1 they ought2 and this would mean
 the cessation of one of the chief means
 of production of individual wealth4    
,we also see why a valuable means for   
the discovery of truth is given by the  
inversion of platitudinous implications4
 ,it may happen that another platitude  
is the result of inversion2 but it is   
the fate of any true remark1 especially 
if it is easy to remember by reason of a
 paradoxical form1 to become a platitude
 in course of time4 ,there are rare     

cases of a platitude remaining          
unrepeated for so long that1 by a       
converse process1 it has become         
paradoxical4 ,such1 for example1 is     
,plato's remark that a lie is less      
important than an error in thought4  ,of
 late years1 a method of disguising     
platitudes as paradoxes has been too    
extensively used by ,mr4 ,g4 ,k4        
,chesterton4 ,the method is as follows4 
,take any proposition p which holds of  
an entity a2 choose p so that it seems  
plausible that p also holds of at least 
two other entities b and c2 call a1 b1  
c1 and any others for which p holds or  
seems to hold1 the class ,a1 and p the  
,a-ness or ,a-ity of ,a2 let d be an    
entity for which p does not hold2 and   
put d among the ,a's when you think that
 nobody is looking4 ,then state your    
paradox3 ,some ,a's do not have ,a-ness4
 ,by further manipulation you can get   
the proposition ,no ,a's have ,a-ness4  
,but it is possible to make a very      
successful coup if ,a is the null-class1

 which has the advantage that           
manipulation is unnecessary4 ,thus1 ,mr4
 ,chesterton1 in his ,orthodoxy put ,a  
for the class of doubters who doubt the 
possibility of logic1 and proved that   
such agnostics refuted themselves--a    
conclusion which seems to have pleased  
many clergymen4  ,in this way1 ,mr4     
,chesterton has been enabled readily to 
write many books and to maintain1 on    
almost every page1 such theses as that  
simplicity is not simple1 heterodoxy is 
not heterodox1 poets are not poetical1  
and so on2 thereby building up the      
gigantic platitude that ,mr4 ,chesterton
 is ,chestertonian4  ,in the chapter on 
,identity we have illustrated the use of
 a case of the principle that any       
proposition implies any true            
proposition4 ,this important principle  
may be called the principle of the      
irrelevant premiss2#ec and is of great  
service in oratory1 because it does not 
matter what the premiss is1 true or     
false4 ,there is a principle of the     

irrelevant conclusion1 but1 except in   
law-courts1 interruptions of meetings1  
and family life1 this is seldom used1   
partly because of the limitation        
involved in the logical impossibility   
for the conclusion to be false if the   
premiss be true1 but chiefly because the
 conclusion is more important than the  
premiss1 being usually a matter of      
prejudice4  ,certain modern logicians1  
such as ,frege1 have found it necessary 
so to extend the meaning of implication 
of q by p that it holds when p is not a 
proposition at all4 ,hitherto1          
politicians1 finding that either        
identical or false propositions are     
sufficient for their needs1 have made no
 use of this principle2 but it is       
obvious that their stock of arguments   
would be vastly increased thereby4      
,logical implication is often an enemy  
of dignity and eloquence4 ,de ,morgan#ed
 relates a tradition of a ,cambridge    
professor who was once asked in a       
mathematical discussion1 ',i suppose you

 will admit that the whole is greater   
than its part8' and who answered1 ',not 
,i1 until ,i see what use you are going 
to make of it4' ,and the care displayed 
by cautious mathematicians like         
,poincare1 ,schoenflies1 ,borel1        
,hobson1 and ,baire in abstaining from  
pushing their arguments to their logical
 conclusions is probably founded on the 
unconscious--but no less                
well-grounded--fear of appearing        
ridiculous if they dealt with such      
extreme cases as the series of all      
ordinal numbers4#ee ,they are1 probably1
 as unconscious of implication as       
,gibbon1 when he remarked that he always
 had a copy of ,horace in his pocket1   
and often in his hand1 was of the       
necessary implication of these          
propositions that his hand was sometimes
 in his pocket4  ,,footnotes3  #ea ,md41
 ,n4 ,s41 vol4 iii41 #ahid1 pp4 #dcf-h4 
,cf4 the discussions by ,w4 ,e4 ,johnson
 7ibid41 p4 #ehc7 and ,russell 7,p4 ,m41
 p4 #ah1 note1 and ,md41 ,n4 ,s41 vol4  

xiv41 #aije1 pp4 #djj-a74  #eb ,the     
inhabitants of ,erewhon punished        
invalids more severely than criminals4  
,in modern times1 one frequently hears  
the statement that crime is a disease2  
and if so1 it is surely false that      
criminals ought to be punished4  #ec    
,irrelevant in a popular sense2 one     
would not say1 speaking loosely1 that   
the fact that ,brutus killed ,caesar    
implies that the sea is salt2 and yet   
this conclusion is implied both by the  
above premiss1 and the premiss that     
,caesar killed ,brutus4 ,cf4 on such    
questions ,venn1 ,s4 ,l41 #bnd ed41 pp4 
#bdj-d4  #ed ,f4 ,l41 p4 #bfd4  #ee ,cf4
 ,chapters ,,xxix and ,,xxxvii4         
,,chapter ,,xx  ,,dignity   ,we have    
seen1 at the end of the preceding       
chapter1 that logical implication is    
often an enemy of dignity4 ,the subject 
of dignity is not usually considered in 
treatises on logic1 but1 as we have     
remarked1 many mathematicians implicitly
 or explicitly seem to fear either that 

the dignity of mathematics will be      
impaired if she follows out conclusions 
logically1 or that only an act of faith 
can save us from the belief that1 if we 
followed out conclusions logically1 we  
should find out something alarming about
 the past1 present1 or future of        
mathematics4  ,thus it seems necessary  
to inquire rather more closely into the 
nature of dignity1 with a view to the   
discovery of whether it is1 as is       
commonly supposed1 a merit in life and  
logic4  ,the chief use of dignity is to 
veil ignorance4 ,thus1 it is well known 
that the most dignified people1 as a    
rule1 are schoolmasters1 and            
schoolmasters are usually so occupied   
with teaching that they have no time to 
learn anything4 ,and because dignity is 
used to hide ignorance1 it is plain that
 impudence is not always the opposite of
 dignity1 but that dignity is sometimes 
impudence4 ,dignity is said to inspire  
respect2 and this may be in part why    
respect for others is an error of       

judgment and self-respect is ridiculous4
  ,self-respect is1 of course1          
self-esteem4 ,william ,james has        
remarked that self-esteem depends1 not  
simply upon our success1 but upon the   
ratio of our success to our pretensions1
 and can therefore be increased by      
diminishing our pretensions4 ,thus if a 
man is successful1 but only then1 can he
 be both ambitious and dignified4 ,james
 also implies that happiness increases  
with self-esteem4 ,likeness of thought  
with one's friends1 then1 does not make 
one happy1 for otherwise a man who      
esteemed himself little would be indeed 
happy4 ,also if a man is unhappy he     
could not1 from our premisses1 by the   
principles of the syllogism and of      
contraposition1 be dignified--a         
conclusion which should be fatal to many
 novelists' heroes4  ,a reflection on   
pessimism to which this discussion gives
 rise is the following3 ,it would appear
 that a man's self-esteem would be      
increased by a conviction of the        

unworthiness of his neighbours4 ,a man1 
therefore1 who thinks that the world and
 all its inhabitants1 except himself1   
are very bad1 should be extremely happy4
 ,in fact1 the effects would hardly be  
distinguishable from those of optimism4 
,and optimism1 as everybody knows1 is a 
state of mind induced by stupidity4     
,,chapter ,,xxi  ,,the ,,synthetic      
,,nature ,,of ,,deduction   ,doubt has  
often been expressed as to whether a    
syllogism can add to our knowledge in   
any way4 ,john ,stuart ,mill and ,henri 
,poincare1 in particular1 held the      
opinion that the conclusion of a        
syllogism is an analytic judgment in the
 sense of ,kant1 and therefore could be 
obtained by the mere dissection of the  
premisses4 ,any one1 then1 who maintains
 that mathematics is founded solely on  
logical principles would appear to      
maintain that mathematics1 in the last  
instance1 reduces to a huge tautology4  
,mill1 in ,chapter ,,iii of ,book ,,ii  
of his ,system of ,logic1 said that it  

must be granted that in every syllogism1
 considered as an argument to prove the 
conclusion1 there is a petitio          
principii4 ,when we say    ,all men are 
mortal1   ,socrates is a man1  therefore
    ,socrates is mortal1  it is         
unanswerably urged by the adversaries of
 the syllogistic theory1 that the       
proposition1 ,socrates is mortal1 is    
presupposed in the more general         
assumption1 ,all men are mortal2 that we
 cannot be assured of the mortality of  
all men unless we are already certain of
 the mortality of every individual man2 
that if it be still doubtful whether    
,socrates1 or any other individual we   
choose to name1 be mortal or not1 the   
same degree of uncertainty must hang    
over the assertion1 ,all men are mortal2
 that the general principle1 instead of 
being given as evidence of the          
particular case1 cannot itself be taken 
for true without exception until every  
shadow of doubt which could affect any  
case comprised with it is dispelled by  

evidence aliunde2 and then what remains 
for the syllogism to prove8 ,that1 in   
short1 no reasoning from general to     
particular can1 as such1 prove anything1
 since from a general principle we      
cannot infer any particulars but those  
which the principle itself assumes as   
known4 ,this doctrine appears to me     
irrefragable4444  ,but it is not        
difficult to see that in certain cases  
at least deduction gives us new         
knowledge4#ef ,if we already know that  
two and two always make four1 and that  
,asquith and ,lloyd ,george are two and 
so are the ,german ,emperor and the     
,crown ,prince1 we can deduce that      
,asquith and ,lloyd ,george and the     
,german ,emperor and the ,crown ,prince 
are four4 ,this is new knowledge1 not   
contained in our premisses1 because the 
general proposition1 two and two are    
four1 never told us there were such     
people as ,asquith and ,lloyd ,george   
and the ,german ,emperor and the ,crown 
,prince1 and the particular premisses   

did not tell us that there were four of 
them1 whereas the particular proposition
 deduced does tell us both these things4
 ,but the newness of the knowledge is   
much less certain if we take the stock  
instance of deduction that is always    
given in books on logic1 namely ,all men
 are mortal2 ,socrates is a man1        
therefore ,socrates is mortal4 ,in this 
case what we really know beyond         
reasonable doubt is that certain men1   
,a1 ,b1 ,c1 were mortal1 since1 in fact1
 they have died4 ,if ,socrates is one of
 these men1 it is foolish to go the     
roundabout way through all men are      
mortal to arrive at the conclusion that 
probably ,socrates is mortal4 ,if       
,socrates is not one of the men on whom 
our induction is based1 we shall still  
do better to argue straight from our ,a1
 ,b1 ,c1 to ,socrates1 than to go round 
by the general proposition1 all men are 
mortal4 ,for the probability that       
,socrates is mortal is greater1 on our  
data1 than the probability that all men 

are mortal4 ,this is obvious1 because if
 all men are mortal1 so is ,socrates2   
but if ,socrates is mortal1 it does not 
follow that all men are mortal4 ,hence  
we shall reach the conclusion that      
,socrates is mortal1 with a greater     
approach to certainty if we make our    
argument purely inductive than if we go 
by way of all men are mortal and then   
use deduction4  ,many years ago there   
appeared1 principally owing to the      
initiative of ,dr4 ,f4 ,c4 ,s4 ,schiller
 of ,oxford1 a comic number of ,mind4   
,the idea was extraordinarily good1 not 
so the execution4 ,a ,german friend of  
,dr4 ,schiller was puzzled by the       
appearance of the advertisements1 which 
were doubtfully humorous4 ,however1 by a
 syllogistic process1 he acquired       
information which was new and useful to 
him1 and thus incidentally refuted      
,mill4 ,presumably he started from the  
title of the magazine 7,mind671 for a   
mark of exclamation seems nearly always 
in ,german to be a sign of an intended  

joke 7including of course the mark after
 the politeness expressed in the first  
sentence of a private letter or a public
 address74 ,there would be1 then1 the   
following syllogism3    ,this is a book 
of would-be jokes 7i4e4 everything in   
this book is a           would-be joke72
   ,this advertisement is in this book2 
,therefore1 this advertisement is a     
would-be joke4  ,thus the syllogism may 
be almost as powerful an agent in the   
detection of humour as ,m4 ,bergson's   
criterion1 to be described in a future  
chapter4#eg  ,,footnotes3  #ef ,the     
following passage is almost word for    
word the same as a passage on pp4 #abc-e
 of ,mr4 ,russell's ,problems of        
,philosophy1 first published in #aiab1 a
 year after ,mr4 ,r99ss99ll's death4 ,it
 is easy hastily to conclude that ,mr4  
,russell was indebted to ,mr4 ,r99ss99ll
 to a greater degree than is usually    
supposed4 ,but an examination of the    
internal evidence leads us to another   
conclusion4 ,the two texts1 it will be  

found1 differ only in the names of the  
,german ,emperor1 the ,crown ,prince and
 the other personages being replaced1 in
 the book of #aiab1 by those of ,messrs4
 ,brown1 ,jones1 ,smith1 and ,robinson4 
,now1 ,mr4 ,russell1 in a new edition of
 his ,problems issued near the beginning
 of the ,european war and before the    
,russian revolution1 substituted the    
,emperor of ,russia for the ,emperor of 
,china of the first edition4 ,hence it  
seems quite likely that ,mr4 ,russell1  
who has always shown a tendency to      
substitute existents for nonentities1   
wrote ,mr4 ,r99ss99ll's                 
notes4--sc,ed4_/sc  #eg ,see ,chapter   
,,xlii4--,,ed4     ,,chapter ,,xxii     
,,the ,,mortality ,,of ,,socrates   ,the
 mortality of ,socrates is so often     
asserted in books on logic that it may  
be as well briefly to consider what it  
means4 ,the phrase ,socrates is mortal  
may be thus defined3 ,there is at least 
one instant t such that t has not to    
,socrates the one-many relation ,r which

 is the converse of the relation 'exists
 at1' and all instants following t have 
not the relation ,r to ,socrates1 and   
there is at least one instant t' such   
that neither t' nor any instant         
preceding t' has the relation ,r to     
,socrates4  ,this definition has many   
merits4 ,in the first place1 no         
assumption is made that ,socrates ever  
lived at all4 ,in the second place1 no  
assumption is made that the instants of 
time form a continuous series4 ,in the  
third place1 no assumption is made as to
 whether ,socrates had a first or last  
moment of his existence4 ,if time be    
indeed a continuous series1 then we can 
easily deduce#eh that there must have   
been either a first moment of his       
non-existence or a last one of his      
existence1 but not both2 just as there  
seems to be either a greatest weight    
that a man can lift or a least weight   
that he cannot lift1 but not both4#ei   
,this may be set forth as follows3 for  
the present we will not concern         

ourselves with evidence for or against  
human immortality2 ,i will merely try to
 present some logical questions which   
persistently arise whenever we think of 
eternal life4 ,one of the greatest      
merits of modern logic is that it has   
allowed us to give precision to such    
problems1 while definitely abandoning   
any pretensions of solving them2 and ,i 
will now apply the logico-analytical    
method to one of the problems of our    
knowledge of the eternal world4#fj  ,we 
will start from the generally accepted  
proposition that all men are mortal4    
,clearly1 if we could know each         
individual man1 and know that he was    
mortal1 that would not enable us to know
 that all men are mortal1 unless we     
knew1 in addition1 that those were all  
the men there are4 ,but we need not here
 assume any such knowledge of general   
propositions2 and1 though most of us    
will admit that the proposition in      
question has great intrinsic            
plausibility1 it is not strictly        

necessary for our present purpose to    
assume anything more than the still more
 probable proposition ,socrates is      
mortal4 ,this last proposition1 quite   
apart from the fact that we have a large
 amount of historical evidence for its  
truth1 has been repeated so often in    
books on logic that it has taken on the 
respectable air of a platitude while    
preserving the character of an          
exceedingly probable truth4 ,the truth  
also results from the fact that it is   
used as the conclusion of a syllogism4  
,for it is a well-known fact that       
syllogisms can only be regarded as      
forming part of a sound education if the
 conclusions are obviously true4 ,the   
use of a syllogism of the form ,all cats
 are ducks and all ducks are mice1      
therefore all cats are mice1 would      
introduce grave doubts into the         
,university of ,oxford as to whether    
logic could any longer be considered as 
a valuable mental training for what are 
amusingly called the learned            

professions4  ,if1 then1 we divide all  
the instants of time1 whether past1     
present1 or future1 into two            
series--those instants at which         
,socrates was alive1 and those instants 
at which he was not alive--and leave out
 of consideration1 for the sake of      
greater simplicity1 all those instants  
before he lived1 we see at once1 by the 
simple application of ,dedekind's       
,axiom1 that1 if ,socrates entered into 
eternal life after his death1 there must
 have been either a last moment of his  
earthly life or a first moment of his   
eternal life1 but not both4  ,logic     
alone can give us no information as to  
which of these cases actually occurred1 
and we are thrown back on to a          
discussion of empirical evidence4 ,it is
 no unusual thing to read of people who 
thought that every moment would be their
 last4 ,in this case it is quite obvious
 that they consequently thought that    
eternity would have no beginning4  ,now 
here we must consider two things3 7#a7  

,it is plainly unsafe to conclude from  
what people think will happen to what   
will happen2 7#b7 even if we could so   
conclude1 it would be unsafe to deduce  
that there was a last moment in the life
 of ,socrates3 we could only make the   
guess plausible1 as we should be using  
the inductive method4  ,there are two   
other pieces of evidence that there is a
 last moment of any earthly existence1  
which we may now briefly consider4 ,that
 this was so was held by ,carlo         
,michaelstaedter2 but since he          
apparently only believed this because he
 wanted1 by attributing a supposed      
ethical value to that moment1 to give   
support to his theory of suicide1 we    
ought not to give great weight to this  
evidence4 ,secondly1 ,thomas ,hobbes    
objected to the principle that a        
quantity may grow less and less         
eternally1 so as at last to be equal to 
another quantity2 or1 which is all one1 
that there is a last in eternity as void
 of sense4 ,now1 the principle meant is 

true1 so that1 although the other       
proposition mentioned by ,hobbes does   
not follow logically from the first1    
there is some evidence that this other  
is true4 ,in fact1 that ,hobbes thought 
that such-and-such a proposition        
followed from another proposition which 
he wrongly believed to be false1 is far 
better evidence for the truth of        
such-and-such a proposition than any we 
have for the truth of most of our most  
cherished beliefs4  ,thirdly1 ,leibniz1 
in a dialogue#fa written on his journey 
of #afgf to visit ,spinoza1 raised the  
question whether the moment at which a  
man dies may be regarded as both the    
last moment at which he is alive and the
 first at which he is dead1 as it must  
be by ,aristotle's theory of continuity4
 ,agreement with this view violates the 
law of contradiction2 denial of it      
implies that two moments can be         
immediately adjacent4 ,by the denial1   
then1 we are led to regard space and    
time as made up of indivisible points   

and moments1 and thus1 since we can draw
 one and only one parallel from any     
point in the diagonal of a square to a  
given side1 the diagonal will contain   
the same 7infinite7 number of points as 
that side1 and will therefore be equal  
to it4 ,in this ,leibniz repeated an    
argument used by the ancient ,arabs1    
,roger ,bacon1 and ,william of ,occam4  
,this ,leibniz considered to be a proof 
that a line cannot be an aggregate of   
points4 ,indeed1 their number would be  
the number of all numbers of the        
greatest possible integer1 which is not4
  ,it does not seem1 further1 that any  
light is thrown on the logical question 
of human mortality or immortality by    
legal decisions4 ,it would appear that  
one can1 legally speaking1 be alive for 
any period less than twenty-four hours  
after one is dead and be dead for any   
period less than twenty-four hours      
before one's death4 ,at least1 according
 to ,salkeld1 i4 #dd1 it was adjudged   
that if one be born the first of        

,february at eleven at night1 and the   
last of ,january in the twenty-first    
year of his age1 at one of the clock in 
the morning1 he makes his will of lands1
 and dies1 it is a good will1 for he was
 then of age4 ,in ,sir ,robert ,howard's
 case 7ibid41 ii4 #fbe7 it was held by  
,chief ,justice ,holt that if ,a be born
 on the third day of ,september2 and on 
the second day of ,september twenty-one 
years afterwards he make his will1 this 
is a good will2 for the law will make no
 fraction of a day1 and by consequence  
he was of age4 ,but it is hardly        
necessary to remark that in this way the
 problem with which we are concerned is 
merely shifted and not solved4 ,for the 
question as to whether there is or is   
not a last moment of a man's life is not
 answered by the decision that he dies  
legally twenty-four hours before or     
after he dies in the usual sense of the 
word1 and the problem arises as to      
whether there is or is not a last moment
 of his legal age4#fb  ,so assuming that

 there was a last moment of ,socrates's 
earthly life1 and consequently no first 
moment of his eternal life1 we see1     
further1 that1 unless the possibility of
 infinite numbers is granted1 it would  
be quite possible for us logically to   
doubt the possibility of an eternal life
 for ,socrates on the same grounds as   
those which led ,zeno to assert that    
motion was impossible and that ,achilles
 could never overtake the ,tortoise4    
,if1 on the other hand1 it be admitted  
that eternity1 at least in the case of  
,socrates1 had a beginning1 these same  
arguments of ,zeno would lead any one   
who denies the possibility of infinite  
number to conclude that ,socrates1 like 
the worm1 can never die4 ,thus is it    
quite plain that the difficulties about 
immortality which meet us at the very   
outset of our inquiry can partly be     
solved only by the help of the theory of
 infinite numbers and partly1 it would  
seem1 not at all4  ,there is another    
difficulty about immortality which is   

quite distinct from this and is         
analogous to another argument of ,zeno4 
,if1 indeed1 all the instants of time be
 divided1 as before1 into the two series
 of instants at which ,socrates was     
alive and instants at which he was not  
alive1 it follows at once that no       
instant of time is not accounted for4   
,at none of these instants1 however1    
does ,socrates die2 obviously he cannot 
die either when he is alive or when he  
is dead4 ,thus it would appear that     
,socrates never died1 and that we ought 
to re-define the term mortal to mean a  
human being who is alive at some moments
 and dead at some4 ,consequently we must
 avoid the very tempting conclusion     
that1 because ,socrates never died1 he  
was therefore immortal4  ,it is very    
important carefully to distinguish      
between the two arguments ,i have just  
set forth4 ,the second argument proves  
quite rigidly that ,socrates and1       
indeed1 anybody else1 never dies1       
whether there is or is not a last moment

 of his life on earth4 ,the first       
argument proves that1 if there is a     
first moment of ,socrates's eternal     
life1 his life on earth never ends4 ,but
 we have seen that we cannot conclude   
that this unending life proves that he  
never is or will be in a state of       
eternity4  ,,footnotes3  #eh ,by        
,dedekind's ,axiom1 ,e4 ,n41 p4 #aa4    
#ei ,m41 vol4 xx41 #aiaj1 pp4 #acd-e4   
#fj ,here1 again1 ,mr4 ,r99ss99ll's work
 seems to anticipate some of ,mr4       
,russell's later work1 e4g4 in ,our     
,knowledge of the ,external ,world as a 
,field for ,scientific ,method in       
,philosophy1 ,chicago and ,london1      
#aiad1 pp4 #c-d1 #ee-f1 et              
passim4--,,ed4  #fa ,pacidius           
,philalethi in ,louis ,couturat1        
,opuscules et ,fragments inedits de     
,leibniz1 ,paris1 #aijc1 pp4 #eid-fbg1  
especially pp4 #eii1 #fja1 #fjh1 #faa4  
,cf4 ,a4 ,e4 ,taylor1 ,hastings'        
,encyclopaedia of ,religion and ,ethics1
 vol4 iv41 ,part #b1 ,edinburgh1 #aiab1 

p4 #if4--;,,ed42 ,robert ,latta1        
,leibniz3 ,the ,monadology and other    
,philosophical ,writings1 ,oxford1      
#ahih1 pp4 #ba ff1 #bi 7note72          
,couturat1 ,la ,logique de ,leibniz     
d'apres des documents inedits1 ,paris1  
#aija1 pp4 #acj1 #acb2 and ,russell1    
,ph4 ,l41 pp4 #ajh-af1 #bdc-i4  #fb ,it 
may be remarked that1 according to ,the 
,times of ,december #bj1 #aiag1 ,mr4    
,justice ,sargant1 in the ,chancery     
,division1 also held that the law did   
not recognize fractions of a day1 and   
that ,lord ,blackburn1 in his decision  
7#i ,app4 ,cas41 #cga1 #cgc7 that a man 
born on the thirteenth of ,may #ahec    
attained the age of twenty-one on the   
thirteenth of ,may #ahgd was not        
speaking strictly4--,,ed4     ,,chapter 
,,xxiii  ,,denoting   ,a concept denotes
 when1 if it occurs in a proposition1   
the proposition is not about the        
concept1 but about a term connected in a
 certain peculiar way with the concept4 
,some people often assert that man is   

mortal1 and yet we never see announced  
in ,the ,times that ,man died on a      
certain day at his villa residence      
,camelot at ,upper ,tooting1#fc nor do  
we hear that ,procrastination was again 
the butt of ,mr4 ,plowden's jokes at    
,marylebone ,police ,court last week4   
,that two phrases may have different    
meanings and the same denotation was    
discovered by ,alice and ,frege4        
,alice#fd observed that the road which  
led to ,tweedledum's house was that     
which led to the house of ,tweedledee2  
and ,frege pointed out that the phrases 
the house to which the road that leads  
to ,tweedledum's house leads and the    
house to which the road that leads to   
,tweedledee's house leads have different
 ,sinn1 but the same ,bedeutung4        
,,footnotes3  #fc ,cf4 ,p4 ,m41 pp4     
#ec-d4  #fd ,see ,appendix ,m4          
,,chapter ,,xxiv  ,,the   ,the word the 
implies existence and uniqueness2 it is 
a mistake to talk of the son of         
,so-and-,so if ,so-and-,so has a fine   

family of ten sons4#fe ,people who refer
 to the ,oxford ,movement imply that    
,oxford only moved once2 and those      
quaint people who say that ,a is quite  
the gentleman imply both the doubtful   
proposition that there is only one      
gentleman in the world1 and the         
indubitably false proposition that he is
 that man4 ,probably ,a is one of those 
persons who add to the confusion in the 
use of the definite article by speaking 
of his wife as the wife4  ,in a certain 
,children's ,hymn ,book one reads3      
,the river vast and small4  ,few would  
deny that there is not more than one    
such river1 but unfortunately it is     
doubtful if there is such a river at    
all4 ,the case is exactly the same with 
the ontological proof of the existence  
of the most perfect being4#ff           
,according to the ,daily ,mail of       
,october #i1 #aijf1 ,judge ,russell     
decided against a claim brought by an   
agent against his company for appointing
 another agent1 the claim being on the  

ground that he was appointed as the     
agent4  ,most people admit that the     
number #b can be added to the number #b 
to give the number #d1 but this is a    
mistake4 ,they concede1 when they use   
the1 that there is only one number #b1  
and yet they imagine that1 when they    
consider it apart as the first term of  
our above sum1 they can find another to 
add to it1 and thereby form the third   
term4 ,the truth is that #b  #b  #d is a
 very misleading equation1 and what we  
really mean by that faultily abbreviated
 statement is more precisely3 ,if x and 
y denote any things which form a class  
,b1 and x' and y' any other things that 
form a class 7,a7 which1 like that of x 
and y1 is a member of the class 7which  
we call #b7 of those classes which have 
a one-one correspondence with ,b 7so    
that any member of ,a corresponds to    
one1 and only one1 member of ,b1 and    
conversely71 the class of all the terms 
of ,a and ,b together is a member of    
that class of classes which1            

analogously1 we call #d4 ,in this1 for  
the sake of shortness1 we have          
introduced abbreviations which should   
not be used in a rigorous logical       
statement4  ,,footnotes3  #fe ,cf4 ,md41
 ,n4 ,s41 vol4 xiv41 #aije1 pp4 #dha1   
#dhd4  #ff ,cf4 ibid41 p4 #dia1 note4   
,,chapter ,,xxv  ,,non-,,entity   ,when 
people say that such-and-such a thing is
 non-existent they usually mean that    
there is not any thing of the kind      
spoken of4 ,venn meant this when he     
described#fg his encounter with what he 
imagined to be a very ingenious         
tradesman3 ,i once had some strawberry  
plants furnished me which the vendor    
admitted would not bear many berries4   
,but he assured me that this did not    
matter1 since they made up in their size
 what they lost in their number4 7,he   
gave me1 in fact1 the hyperbolic        
formula1 xy  c1 to connect the number   
and magnitude47 ,when summer came1 no   
fruit whatever appeared4 ,i saw that it 
would be no use to complain1 because the

 man would urge that the size of the    
non-existent berry was infinite1 which  
,i could not see my way to disprove4 ,i 
had forgotten to bar zero values of     
either variable4  ,it is to be regretted
 that this useful note was omitted in   
the second edition of ,venn's book2 one 
can imagine that it might have protected
 ,mr4 ,mac,coll and ,herr ,meinong 7who 
believed1 unlike ,alice in what may be  
called her first theory1#fh in round    
squares and fabulous monsters7 against  
the dishonest practices of traders who  
were too ready with promises4 ,for the  
death-blow to this kind of trade was not
 given until #aije1 when ,mr4 ,russell  
published his article ,on ,denoting1#fi 
and took up the position of the ,white  
,king in opposition to ,alice's later   
assertions4#gj  ,venn's experience      
illustrates another characteristic of   
mathematical logic4 ,it is necessary1 in
 order to make our arguments conclusive1
 to devote great care to the elimination
 of difficulties which rarely occur4    

,the ,white ,knight--who was like ,boole
 in being a pioneer of mathematical     
logic in this way1 and yet seems to have
 held1 like ,boole1 those philosophical 
opinions which would base logic on      
psychology--recognized the necessity of 
taking precautions against any unusual  
appearance of mice on a horse's back4#ga
  ,,footnotes3  #fg ,s4 ,l41 #ahha1 p4  
#cci1 note4  #fh ,see ,appendix ,n4  #fi
 ,md41 ,n4 ,s41 vol4 xiv41 ,october     
#aije1 pp4 #dgi-ic4  #gj ,see ,appendix 
,n4  #ga ,see ,appendix ,o4             
,,chapter ,,xxvi  ,,is   ,is has four   
perfectly distinct meanings in ,english1
 besides misuses of the word4 ,among the
 misuses1 perhaps the most important are
 those referred to by ,de ,morgan3#gb   
444 ,we say 'murder is death to the     
perpetrator' where the copula is brings2
 'two and two are four1' the copula     
being 'have the value of1' etc4         
,schroeder#gc quite satisfactorily      
pointed out the well-known distinction  
between an is where subject and         

predicate can be interchanged 7such as3 
the class whose members are ,shem1 ,ham 
and ,japhet is the class of the sons of 
,noah7 and an is or are where they      
cannot 7such as3 ,englishmen are        
,britons71 but failed to see#gd the more
 important distinction 7made by ,peano7 
of is in the sense of is a member of4   
,if ,englishmen are ,britons1 and       
,britons are civilized people1 it       
follows that ,englishmen are civilized  
people2 but1 though the ,harmsworth     
,encyclopaedia is a member of the class 
,book 7of one or more volumes71 and this
 class is the member of a class ,a of   
which it is the only member1 yet the    
,harmsworth ,encyclopaedia is not a     
member of ,a1 for it is not true that it
 is the whole class of books2 and such a
 statement would not even be made except
 possibly in the form of an             
advertisement4  ,the fourth meaning of  
is is exists2 it is in certain rare     
moods a matter for regret that there are
 difficulties in the way of using one   

word to denote four different things4   
,for1 if there were not1 we might prove 
the existence of any thing we please by 
making it the subject of a proposition1 
and thereby earn the gratitude of       
theologians4  ,,footnotes3  #gb ,f4 ,l41
 p4 #bfh4  #gc ,a4 d4 ,l41 i4 pp4 #abg  
sqq4  #gd ,ibid41 vol4 ii4 pp4 #dfa1    
#eig4     ,,chapter ,,xxvii  ,,and ,,and
 ,,or   ,when1 with ,boole1 alternatives
 7,a1 ,b7 are considered as mutually    
exclusive1 logical addition may be      
described as the process of taking ,a   
and ,b or ,a or ,b4 ,it is a great and  
rare convenience to have two terms for  
denoting the same thing3 commonly1      
people denote several things by the same
 term1 and only the ,germans have the   
privilege of referring to1 say1         
continuity as ,stetigkeit or            
,kontinuierlichkeit4 ,but ,jevons#ge    
quoted ,milton1 ,shakespeare1 and       
,darwin to prove that alternatives are  
not exclusive1 and so attained first to 
recognized views by arguments which were

 plainly irrelevant4  ,of course1 and is
 often used as the sign of logical      
addition3 thus one may speak of one's   
brothers and sisters1 without being     
understood to mean the null-class 7as   
should be the case71 or pray for one's  
relations and friends1 without being    
sure that one's prayer would be         
answered1--as it certainly would if one 
meant to pray for the null-class1 this  
being the class indicated4 ,and a word  
like while is often used for a logical  
addition1 when exclusiveness of the     
alternatives is almost implied4 ,thus1 a
 reviewer in ,mind1#gf noticing the     
translation of ,mach's ,popular         
,scientific ,lectures into ,american1   
said of the lectures that3 ,most of them
 will be familiar 444 to epistemologists
 and experimental psychologists3 while  
the remainder1 which deal with physical 
questions1 are well worth reading4 ,the 
reader has the impression1 probably     
given unintentionally1 that ,professor  
,mach's epistemological and             

psychological lectures are not1 in the  
reviewer's opinion1 worth reading4      
,,footnotes3  #ge ,pure ,logic 4441     
,london1 #ahfd1 pp4 #gf-i4 ,cf4 ,venn1  
,s4 ,l41 #bnd ed41 pp4 #dj-h4  #gf ,n4  
,s41 vol4 iv4 p4 #bfa4     ,,chapter    
,,xxviii  ,,the ,,conversion ,,of       
,,relations   ,the ,conversion of       
,relations does not mean what it might  
be supposed to mean2 it has nothing to  
do with what ,kant called the wholesome 
art of persuasion4 ,what concerns us    
here is the convertibility of a logical 
relation4 ,if ,a has a certain relation 
,r to ,b1 the relation of ,b to ,a1     
which may be denoted by v,r1 is called  
the converse of ,r4 ,as ,de ,morgan#gg  
remarked1 this conversion may sometimes 
present difficulties4 ,the following is 
,de ,morgan's example3  ,teacher3 ',now1
 boys1 ,shem1 ,ham and ,japheth were    
,noah's sons2 who was the father of     
,shem1 ,ham and ,japheth8' ,no answer4  
,teacher3 ',boys1 you know ,mr4 ,smith1 
the carpenter1 opposite2 has he any     

sons8'  ,boys3 ',oh6 yes1 sir6 there's  
,bill and ,ben4'  ,teacher3 ',and who is
 the father of ,bill and ,ben ,smith8'  
,boys3 ',why1 ,mr4 ,smith1 to be sure4' 
,teacher3 ',well1 then1 once more1      
,shem1 ,ham and ,japheth were ,noah's   
sons2 who was the father of ,shem1 ,ham 
and ,japheth8'  ,a long pause2 at last a
 boy1 indignant at what he thought the  
attempted trick1 cried out3 ',it        
couldn't have been ,mr4 ,smith4' ,these 
boys had never converted the relation of
 father and son4444  ,,footnotes3  #gg  
,trans4 ,camb4 ,phil4 ,soc41 vol4 x41   
#ahfd1 part ii41 note on page #ccd4     
,,chapter ,,xxix  ,,previous            
,,philosophical ,,theories ,,of         
,,mathematics   ,mathematicians usually 
try to found mathematics on two         
principles3#gh one is the principle of  
confusion between the sign and the thing
 signified 7they call this principle the
 foundation-stone of the formal theory71
 and the other is the ,principle of the 
,identity of ,discernibles 7which they  

call the principle of the permanence of 
equivalent forms74  ,but the truth is   
that if we set sail on a voyage of      
discovery with ,logic alone at the helm1
 we must either throw such principles as
 the identity of those conceptions which
 have in common the properties that     
interest us and the principle of        
permanence overboard1 or1 if we do not  
like to act in such a way to old        
companions with whom we are so familiar 
that we can hardly feel contempt for    
them1 at least recognize them clearly as
 having no logical validity and merely  
as psychological principles1 and reduce 
them to the humble rank of stewards1 to 
minister to our human weaknesses on the 
voyage4 ,and then1 if we adopt the wise 
policy of keeping our axioms down to the
 minimum number1 we must refrain from   
creating or thinking that we are        
creating new numbers to fill up gaps    
among the older ones1 and thence        
recognize that our rational numbers are 
not particular cases of real numbers1   

and so on4  ,we thus get a world of     
conceptions which looks1 and is1 very   
different from that which ordinary      
mathematicians think they see2 and      
perhaps this is the reason why some     
mathematicians of great eminence1 such  
as ,hilbert and ,poincare1 have produced
 such absurd discussions on the         
fundamental principles of               
mathematics1#gi showing once more the   
truth of the not quite original remark  
of ,aunt ,jane1 who    444 observed1 the
 second time   ,she tumbled off a 'bus3 
,the step is short from the sublime     
,to the ridiculous4  ,in their readiness
 to consider many different things as   
one thing--to consider1 for example1 the
 ratio #b3a as the same thing as the    
cardinal number #b--such mathematicians 
as ,peacock1 ,hankel1 and ,schubert were
 forestalled by the ,pigeon1 who thought
 that ,alice and the ,serpent were the  
same creature1 because both had long    
necks and ate eggs4#hj ,it is1 however1 
doubtful whether the ,pigeon would have 

followed the example of the             
mathematicians just mentioned so far as 
to embrace the creed of nominalism and  
so to feel no difficulty in subtracting 
from zero--a difficulty which was       
pointed out with great acuteness by the 
,hatter#ha and modern mathematical      
logicians4  ,,footnotes3  #gh ,these    
principles1 after many attempts to state
 them by ,peacock1 the ,red and the     
,white ,queen 7see ,appendix ,p71       
,hankel1 ,schroeder1 and ,schubert had  
been made1 were first precisely         
formulated by ,frege in ,z4 ,s42 cf4    
also ,chapter ,,vii4  #gi ,see          
,couturat1 ,r4 ,m4 ,m41 vol4 xiv41      
,march1 #aijf1 pp4 #bjh-ej1 and         
,russell1 ibid41 ,september1 #aijf1 pp4 
#fbg-cd4  #hj ,see ,appendix ,p4  #ha   
,see ibid4     ,,chapter ,,xxx  ,,finite
 ,,and ,,infinite   ,i was once shown a 
statement made by an eminent            
mathematician of ,cambridge from which  
one would conclude that this            
mathematician thought that finite       

distances became infinite when they were
 great enough4 ,in one of those         
splendidly printed books1 bound in blue1
 published by the ,university ,press1   
and sold at about a guinea as a guide to
 some advanced branch of pure           
mathematics1 one may read1 even in the  
second edition published in #aijj1 the  
words3 ,representation of a complex     
variable on a plane is obviously more   
effective for points at a finite        
distance from the origin than for points
 at a very great distance4  ,plainly    
some of the points at a very great      
distance are at a finite distance1 for  
the same author mentions that ,neumann's
 sphere for representing the positions  
of points on a plane has the advantage  
444 of exhibiting the uniqueness of z   
infinity symbol as a value of the       
variable4     ,,chapter ,,xxxi  ,,the   
,,mathematical ,,attainments ,,of       
,,tristram ,,shandy   ,tristram         
,shandy#hb said that his father was     
sometimes a gainer by misfortune2 for if

 the pleasure of haranguing about it was
 as ten1 and the misfortune itself only 
as five1 he gained half in half1 and was
 well off again as if the misfortune had
 never happened4  ,suppose that the unit
 7arbitrary7 of pleasure is denoted by  
,a1 ,tristram ,shandy1 by neglecting1 in
 this ethical discussion1 to introduce  
negative quantities 7,kant's pamphlet   
advocating this introduction into       
philosophy was made subsequently#hc71   
apparently made #ae;,a to result1 and   
this can hardly be maintained to be the 
half of #aj;,a4 ,it is possible1        
however1 that ,tristram ,shandy         
succeeded in proving the apparently     
paradoxical equation    #ae;,a  #e;,a   
by remarking that the axiom the whole is
 greater than the part does not always  
hold4 ,this remark follows at once from 
what ,mr4 ,russell#hd has called ,the   
,paradox of ,tristram ,shandy4 ,this    
paradox is described by ,mr4 ,russell as
 follows3  ,tristram ,shandy1 as we     
know1 took two years writing the history

 of the first two days of his life1 and 
lamented that1 at this rate1 material   
would accumulate faster than he could   
deal with it1 so that he could never    
come to an end4 ,now ,i maintain that1  
if he had lived for ever1 and not       
wearied of his task1 then1 even if his  
life had continued as eventfully as it  
began1 no part of his biography would   
have remained unwritten4  ,this paradox 
is strictly correlative to the          
well-known paradox of ,zeno about       
,achilles and the ,tortoise4#he ,the    
,achilles proves that two variables in a
 continuous series1 which approach      
equality from the same side1 cannot ever
 have a common limit3 the ,tristram     
,shandy proves that two variables which 
start from a common term1 and proceed in
 the same direction1 but diverge more   
and more1 may yet determine the same    
limiting class 7which1 however1 is not  
necessarily a segment1 because segments 
were defined as having terms beyond     
them74 ,the ,achilles assumes that whole

 and part cannot be similar1 and deduces
 a paradox2 the other1 starting from a  
platitude1 deduces that whole and part  
may be similar4 ,for common-sense1 it   
must be confessed that it is a most     
unfortunate state of things4 ,and ,mr4  
,russell considers that1 in the face of 
proofs1 it ought to commit suicide in   
despair4  ,now1 ,i suggest the extremely
 unlikely possibility that ,tristram    
,shandy1 by reflection on his own life  
and literary labours1 was led to the    
correct course of accepting the paradox 
which resulted from this reflection and 
rejecting the ,achilles4 ,thus1 he      
concluded that an infinite whole may be 
similar 7or1 in ,cantor's terminology1  
equivalent7 to a proper part of itself1 
and hence1 by a confusion of similarity 
with identity 7or equivalence with      
equality7 which he shares with some     
subsequent philosophers1#hf that a whole
 may be equal to a proper part of       
itself4 ,if ,a is an infinite class1 it 
is not difficult to see that we can have

    #aj;,a  #e;,a4  ,in this way many   
have avoided an opinion which rests on  
no better foundation than that formerly 
entertained by the inductive            
philosophers of ,central ,africa1 that  
all men are black4#hg  ,,footnotes3  #hb
 ,cf4 a letter of ,de ,morgan in ,mrs4  
,de ,morgan's ,memoir of ,augustus ,de  
,morgan1 p4 #cbd4  #hc ,kant's tract was
 published in #agfc1 while ,tristram    
,shandy was published in #agfj4  #hd ,p4
 ,m41 pp4 #ceh-i ,cf4 ,m41 vol4 xxii41  
,january #aiab1 p4 #ahg4--;,,ed4  #he   
,cf4 ,p4 ,m41 pp4 #cej1 #ceh-i2 ,m41    
vol4 xxii41 #aiab1 p4 #aeg4  #hf ,cf4   
for example1 ,cosmo ,guastella1 ,dell'  
infinito1 ,palermo1 #aiab4--;,,ed4  #hg 
,cf4 ,russell1 ,p4 ,m41 p4 #cfj4        
,,chapter ,,xxxii  ,,the ,,hardships    
,,of ,a ,,man ,,with ,,an ,,unlimited   
,,income   ,i once heard a man refer to 
his income as limited1 in order to      
illustrate the hardship of a class of   
men1 of which he of course was one1 in  
having to pay a somewhat high           

income-tax4 ,it is obvious that this man
 spoke enviously1 and consequently      
admitted the existence of more          
fortunately placed individuals who had  
unlimited incomes4 ,a little reflection 
would have shown the man that he was not
 taking up a paradoxical attitude4 ,a   
paradoxical attitude is of course the   
assertion of one or more propositions of
 which the truth cannot be perceived by 
a philosopher--and particularly an      
idealist--and can be perceived by a     
logician and occasionally1 but not      
always1 by a man of common-sense4 ,such 
propositions are3 ,the cat is hungry1   
,columbus discovered ,america1 and ,a   
thing which is always at rest may move  
from the position ,a to the different   
position ,b4  ,now1 if a man had an     
unlimited income1 it is an immediate    
inference that1 however low income-tax  
might be1 he would have to pay annually 
to the ,exchequer of his nation a sum   
equal in value to his whole income4     
,further1 if his income was derived from

 a capital invested at a finite rate of 
interest 7as is usual71 the annual      
payments of income-tax would each be    
equal in value to the man's whole       
capital4 ,if1 then1 the man with an     
unlimited income chose to be            
discontented1 he would be sure of a     
sympathetic audience among philosophers 
and business acquaintances2 but         
discontent could not last long1 for the 
thought of the difficulties he was      
putting in the way of the ,chancellor of
 the ,exchequer1 who would find the     
drawing up of his budget most puzzling1 
would be amusing4 ,again1 the discovery 
that1 after paying an infinite          
income-tax1 the income would be quite   
undiminished1 would obviously afford    
satisfaction1 though perhaps the        
satisfaction might be mixed with a      
slight uneasiness as to any action the  
,commissioners of ,income-,tax might    
take in view of this fact4  ,a problem  
of a wholly different nature is         
connected with the possible purchase by 

the man with an unlimited income of an  
enumerable infinity of pairs of boots4  
,if he wished to prove that he had an   
even number of boots1 it would be easy  
if right boots were distinguishable from
 left ones1 but if the man were a       
faddist of such a kind that he insisted 
that his left boots should not be made  
in any way differently from his right   
ones1 it would not be possible for him  
to prove the theorem mentioned unless he
 assumed what is known as the           
multiplicative axiom4 ,in fact this     
axiom shows that it is legitimate to    
pick out an infinite succession of      
members of an infinite class in an      
arbitrary way4 ,in the case of the pairs
 of boots1 each pair contains two       
members1 and if there is no means of    
distinguishing between them1 when we    
wish to pick out one of them for each of
 the infinity of pairs1 we cannot say   
which ones we mean to pick out unless we
 assume1 by means of the above axiom1   
that a particularized member can always 

be found even with things of each of    
which it can be said that1 like ,private
 ,james in the ,bab ,ballads1    ,no    
characteristic trait had he   ,of any   
distinctive kind4  ,however1 a solution 
of the puzzle was given by ,dr4 ,denes  
,koenig of ,budapest4 ,you first prove  
that there are points in space such     
that1 if ,p is one of them1 not more    
than a finite number of pairs of boots  
are such that each centre of mass of the
 two members of a pair is equidistant   
from ,p4 ,taking a point ,p of this     
sort1 select from each pair the boot    
whose centre of mass is nearest ,p4     
7,there may be a finite number of pairs 
left over1 but they can be dealt with   
arbitrarily47  ,another form of the     
problem is as follows4 ,every time the  
man bought a pair of boots he also      
bought a pair of socks to go with it2 he
 had an enumerable infinity of pairs of 
each1 and the problem is to prove that  
he had as many boots as he had socks4   
,in this case the boots1 we will        

suppose1 can be divided into right and  
left1 but the socks cannot4 ,thus there 
are an enumerable infinity of boots1 but
 the number of the socks cannot be      
determined without admitting the axiom  
mentioned above4 ,a further difficulty  
might arise if the owner of the boots   
and socks lost one leg in some accident1
 and told his butler to give away half  
his socks4 ,naturally the butler would  
find great logical difficulties in so   
doing1 and it would seem to be an       
interesting ethical problem whether he  
should be dismissed from his situation  
for failing to prove the multiplicative 
axiom4 ,again1 if the butler stole a    
pair of boots1 the millionaire would    
have as many pairs as before1 but might 
have fewer boots4 ,there is as yet no   
evidence that the number of his boots is
 equal to or greater than the number of 
pairs4     ,,chapter ,,xxxiii  ,,the    
,,relations ,,of ,,magnitude ,,of       
,,cardinal ,,numbers   ,the theorems of 
cardinal arithmetic are frequently used 

in ordinary conversation4 ,what is known
 as the ,schroeder-,bernstein theorem   
was used1 long before ,bernstein or     
,schroeder1 by ,edward ,thurlow1        
afterward the law-lord ,lord ,thurlow1  
when an undergraduate of ,caius         
,college1 ,cambridge4 ,thurlow was      
rebuked for idleness by the ,master1 who
 said to him3 ,whenever ,i look out of  
the window1 ,mr4 ,thurlow1 ,i see you   
crossing the ,court4 ,the provost thus  
asserted a one-one correspondence       
between the class ,a of his acts of     
looking out of the window and a part of 
the class ,b of ,thurlow's acts of      
crossing the ,court4 ,thurlow asserted  
in reply a one-one correspondence       
between ,b and a part of ,a3 ,whenever  
,i cross the ,court ,i see you looking  
out of the window4 ,the                 
,schroeder-,bernstein theorem1 then1    
allows us to conclude that there is a   
one-one correspondence between the      
classes ,a and ,b4 ,that ,a and ,b were 
finite classes is not the fault of the  

,master or ,thurlow2 nor is it relevant 
logically4     ,,chapter ,,xxxiv  ,,the 
,,unknowable   ,according to ,mr4 ,s4   
,n4 ,gupta1#hh the first thing that     
every student of ,hindu logic has to    
learn when he is said to begin the study
 of inference is that all ,h is ,s is   
not always equivalent to ,no ,h is not  
,s4 ,the latter proposition is an       
absurdity when ,s is ,kebalanvayi1 i4e4 
covers the whole sphere of thought and  
existence4444 ',knowable' and           
',nameable' are among the examples of   
,kebalanvayi terms4 ,if you say there is
 a thing not-knowable1 how do you know  
it8 ,if you say there is a thing        
not-nameable1 you must point that out1  
i4e4 somehow name it4 ,thus you         
contradict yourself4  ,mr4 ,herbert     
,spencer's doctrine of the ,unknowable  
gives rise to some amusing thoughts4 ,to
 state that all knowledge of such and   
such a thing is above a certain person's
 intelligence is not self-contradictory1
 but merely rude3 to state that all     

knowledge of a certain thing is above   
all possible human intelligence is      
nonsense1 in spite of its modest1       
platitudinous appearance4 ,for the      
statement seems to show that we do know 
something of it1 viz4 that it is        
unknowable4  ,to the last 7#aijj7       
edition of ,first ,principles was added 
a ,postscript to ,part ,i1 in which the 
justice of this simple and well-known   
criticism as to the contradiction       
involved in speaking of an ,unknowable1 
which had been often made during the    
forty odd years in which the various    
editions had been on the market1 was    
grudgingly acknowledged as follows3#hi  
,it is doubtless true that saying what a
 thing is not1 is1 in some measure1     
saying what it is2444 ,hence it cannot  
be denied that to affirm of the         
,ultimate ,reality that it is unknowable
 is1 in a remote way1 to assert some    
knowledge of it1 and therefore involves 
a contradiction4  ,the ,postscript      
reminds one of the postscript to a      

certain ,irishman's letter4 ,this       
,irishman1 missing his razors after his 
return from a visit to a friend1 wrote  
to his friend1 giving precise directions
 where to look for the missing razors2  
but1 before posting the letter1 added a 
postscript to the effect that he had    
found the razors4  ,one is tempted to   
inquire1 analogously1 what might be1 in 
view of the ,postscript1 the point of   
much of ,spencer's ,part ,i4 ,it is1 to 
use ,de ,morgan's#ij description of the 
arguments of some who maintain that we  
can know nothing about infinity1 of the 
same force as that of the man who       
answered the question how long he had   
been deaf and dumb4  ,but the best part 
of the joke against ,mr4 ,spencer is    
that he1 as we shall see in ,chapter    
,,xxxviii1 was refuted by a fallacious  
argument1 and thus mistakenly asserted  
the validity of the refutation of       
remarks which happen to be unsound4     
,the analogy of the contradiction of    
,burali-,forti with the contradiction   

involved in the notion of an unknowable 
may be set forth as follows4 ,if ,a     
should say to ,b3 ,i know things which  
you never by any possibility can know1  
he may be speaking the truth4 ,in the   
same way1 ,greek3 o may be said1 without
 contradiction1 to transcend all the    
finite integers4 ,but if some one else1 
,c1 should say3 ,there are some things  
which no human being can ever know      
anything about1 he is talking           
nonsense4#ia ,and in the same way if we 
succeeded in imagining a number which   
transcends all numbers1 we have         
succeeded in imagining the absurdity of 
a number which transcends itself4  ,all 
the paradoxes of logic 7or the theory of
 aggregates7 are analogous to the       
difficulty arising from a man's         
statement3 ,i am lying4#ib ,in fact1 if 
this is true1 it is false1 and vice     
versa4 ,if such a statement is spread   
out a little1 it becomes an amusing hoax
 or an epigram4 ,thus1 one may present  
to a friend a card bearing on both sides

 the words3 ,the statement on the other 
side of this card is false2 while the   
first of the epigrams derived from this 
principle seems to have been written by 
a ,greek satirist3#ic    ,lerians are   
bad2 not some bad and some not2   ,but  
all2 there's not a ,lerian in the lot1  
,save ,procles1 that you could a good   
man call2--   ,and ,procles--is a       
,lerian after all4  ,this is the        
original of a well-known epigram by     
,porson1 who remarked that all ,germans 
are ignorant of ,greek metres1    ,all1 
save only ,hermann2--   ,and ,hermann's 
a ,german4  ,,footnotes3  #hh ,md41 ,n4 
,s41 vol4 iv41 #ahie1 p4 #afh4  #hi     
,first ,principles1 #fth ed41 #aijj1 pp4
 #ajg-aj4 ,the first edition was        
published in #ahfb4  #ij ,note on p4 #f 
of his paper3 ,on ,infinity2 and on the 
,sign of ,equality1 ,trans4 ,camb4      
,phil4 ,soc41 vol4 xi41 part i41 pp4    
#a-de 7read ,may #af1 #ahfd74  #ia ,the 
assertion of the finitude of a man's    
mind appears to be nonsense2 both       

because1 if we say that the mind of man 
is limited we tacitly postulate an      
unknowable1 and because1 even if the    
human mind were finite1 there is no more
 reason against its conceiving the      
infinite than there is for a mind to be 
blue in order to conceive a pair of blue
 eyes 7cf4 ,de ,morgan1 loc4 cit474  #ib
 ,russell1 ,r4 ,m4 ,m41 vol4 xiv41      
,september #aijf1 pp4 #fcb-c1 #fdj-d4   
#ic ,the ,greek ,anthology1 by ,lord    
,neaves 7,ancient ,classics for ,english
 ,readers71 ,edinburgh and ,london1     
#ahig1 p4 #aid4     ,,chapter ,,xxxv    
,,mr4 ,,spencer1 ,,the ,,athanasian     
,,creed ,,and ,,the ,,articles   ,when1 
in what ,i believe is misleadingly known
 as ,the ,athanasian ,creed1 people say 
,the ,father incomprehensible1 and so   
on1 they are not falling into the same  
error as ,mr4 ,spencer1 for the ,latin  
equivalent for incomprehensible is      
merely immensus1 and ,bishop ,hilsey    
translated it more correctly as         
immeasurable4#id ,it is a regrettable   

fact that ,dr4 ,blunt1#ie in his        
mistaken modesty1 has added a note to   
this passage that3 ,yet it is true that 
a meaning not intended in the ,creed has
 developed itself through this change of
 language1 for the ,nature of ,god is as
 far beyond the grasp of the mind as it 
is beyond the possibility of being      
contained within local bounds4  ,mr4    
,spencer seems no happier when we       
compare his statements with those in the
 ,anglican ,articles of ,religion4      
,there ,god is never referred to as     
infinite4 ,it is true that ,his power   
and goodness are so referred to2 but    
this deficiency was presumably brought  
about intentionally1 so that faith might
 gain in meaning as time went on4       
,,footnotes3  #id ,a4 ,c4 ,p41 p4 #bag4 
#ie ,ibid41 p4 #bah4     ,,chapter      
,,xxxvi  ,,the ,,humour ,,of            
,,mathematicians   ,brahmagupta's       
problem#if appears to be the earliest   
instance of a kind of joke which has    
been much used by mathematicians4 ,for  

the sake of giving a certain            
picturesqueness to the data of problems1
 and so to excite that sort of interest 
which is partly expressed by a smile1   
mathematicians have got into the habit  
of talking1 for example1 of monkeys in  
the form of geometrical points climbing 
up massless ropes4 ,professor ,p4       
,staeckel#ig truly remarked that        
physiological mechanics--the mechanics  
of bones1 muscles1 and so on--is wholly 
different from this4 ,there was once a  
lecturer on mathematics at ,cambridge   
who used yearly to propound to his      
pupils a problem in rigid dynamics which
 related to the motion of a garden      
roller supposed to be without mass or   
friction1 when a heavy and perfectly    
rough insect walked round the interior  
of it in the direction of normal        
rolling4  ,hitherto this has been the   
only mathematical outlet for the humour 
of mathematicians2 and those who really 
had the interests of mathematics at     
heart saw with alarm the growing        

tendency towards scholasticism in       
mathematical jokes4 ,fortunately the    
discovery of logic by some              
mathematicians has removed this danger4 
,still to many mathematicians logic is  
still unknown1 and to them--to          
,professor ,a4 ,schoenflies for         
example--modern mathematics1 owing to   
its alliance with logic1 appears to be  
sinking into scholasticism4 ,it is true 
that the word scholasticism is not used 
by ,professor ,schoenflies in any       
intentionally precise signification1 but
 merely as a vague epithet of           
disapproval1 as the word socialism is   
used by the ordinary philistine1 and    
this would certainly serve as a         
sufficient excuse4 ,but no excuse is    
needed3 these opinions are themselves a 
source of mathematical jokes4           
,,footnotes3  #if ,see ,chapter ,,xii4  
#ig ,encykl4 der math4 ,wiss41 vol4 iv41
 part i41 p4 #dgd4     ,,chapter        
,,xxxvii  ,,the ,,paradoxes ,,of ,,logic
   ,we have already#ih referred to the  

contempt shown by some mathematicians   
for exact thought1 which they condemn   
under the name of scholasticism4 ,an    
example of this is given by ,schoenflies
 in the second part of his publication  
usually known as the ,bericht ueber     
,mengenlehre4#ii ,here#ajj a battle-cry 
in italics--    ,against all            
resignation1 but also against all       
scholasticism6--  found utterance4      
,later on1 ,schoenflies#aja became      
bolder and adopted a more personal      
battle-cry1 also in italics1 and with a 
whole line to itself3    ,for ,cantorism
 but against ,russellism6  ,cantorism   
means the theory of transfinite         
aggregates and numbers erected for the  
most part by ,georg ,cantor4 ,shortly   
speaking1 the great sin of ,russellism  
is to have gone too far in the chain of 
logical deduction for many              
mathematicians1 who were perhaps1 like  
,schoenflies1#ajb blinded by their      
rather uncritical love of mathematics4  
,thus it comes about that               

,schoenflies#ajc denounces ,russellism  
as scholastic and unhealthy4 ,this queer
 blend of qualities would surely arouse 
the curiosity of the most blase as to   
what strange thing ,russellism must     
be4#ajd  ,schoenflies#aje said that some
 mathematicians attributed to the       
logical paradoxes which have given      
,russell so much trouble to clear up1   
especially to those that are            
artificially constructed1 a significance
 that they do not have4 ,yet no grounds 
were given for this assertion1 from     
which it might be concluded that the    
rigid examination of any concept was    
unimportant4 ,the paradoxes are simply  
the necessary results of certain logical
 views which are currently held1 which  
views do not1 except when they are      
examined rather closely1 appear to      
contain any difficulty4 ,the            
contradiction is not felt1 as it        
happens1 by people who confine their    
attention to the first few              
number-classes of ,cantor1 and this     

seems to have given rise to the opinion1
 which it is a little surprising to find
 that some still hold1 that cases not   
usually met with1 though falling under  
the same concept as those usually met   
with1 are of little importance4 ,one    
might just as well maintain that        
continuous but not differentiable       
functions are unimportant because they  
are artificially constructed--a term    
which ,i suppose means that they do not 
present themselves when unasked for4    
,rather should we say that it is by the 
discovery and investigation of such     
cases that the concept in question can  
alone be judged1 and the validity of    
certain theorems--if they are           
valid--conclusively proved4 ,that this  
has been done1 chiefly by the work of   
,russell1 is simply a fact2 that this   
work has been and is misunderstood by   
many#ajf is regrettable for this reason1
 among others1 that it proves that1 at  
the present time1 as in the days in     
which ,gulliver's ,travels were written1

 some mathematicians are bad            
reasoners4#ajg  ,nearly all             
mathematicians agreed that the way to   
solve these paradoxes was simply not to 
mention them2 but there was some        
divergence of opinion as to how they    
were to be unmentioned4 ,it was clearly 
unsatisfactory merely not to mention    
them4 ,thus ,poincare was apparently of 
opinion that the best way of avoiding   
such awkward subjects was to mention    
that they were not to be mentioned4     
,but#ajh one might as well1 in talking  
to a man with a long nose1 say3 ',when  
,i speak of noses1 ,i except such as are
 inordinately long1' which would not be 
a very successful effort to avoid a     
painful topic4  ,schoenflies1 in his    
paper of #aiaa mentioned above1 adopted 
the convenient plan of referring these  
logical difficulties at the root of     
mathematics to a department of knowledge
 which he called philosophy4 ,he        
said#aji of the theory of aggregates    
that though born of the acuteness of the

 mathematical spirit1 it has gradually  
fallen into philosophical ways1 and has 
lost to some extent the compelling force
 which dwells in the mathematical       
process of conclusion4  ,the majority of
 mathematicians have followed           
,schoenflies rather than ,poincare1 and 
have thus adopted tactics rather like   
those of the ,march ,hare and the       
,gryphon1#aaj who promptly changed the  
subject when ,alice raised awkward      
questions4 ,indeed1 the process of the  
first of these creatures of a child's   
dream is rather preferable to that of   
,schoenflies4 ,the ,march ,hare refused 
to discuss the subject because he was   
bored when difficulties arose4          
,schoenflies would not say that he was  
bored--he professed interest in         
philosophical matters1 but simply called
 the logical continuation of a subject  
by another name when he did not wish to 
discuss the continuation1 and thus      
implied that he had discussed the whole 
subject4 ,further1 ,schoenflies would   

not apparently admit that the one method
 of logic could be applied to the       
solution of both mathematical and       
philosophical problems1 in so far as    
these problems are soluble at all2 but  
the ,march ,hare1 shortly before the    
remark we have just quoted1 rightly     
showed great astonishment that butter   
did not help to cure both hunger and    
watches that would not go4#aaa ,the     
judgment of ,schoenflies by which       
certain apparently mathematical         
questions were condemned as             
philosophical1 rested on grounds as     
flimsy as those in the ,dreyfus ,case1  
or the ,trial in ,wonderland4#aab       
,,footnotes3  #ih ,chapters ,,vii and   
,,xxxvi4  #ii ,die ,entwickelung der    
,lehre von den ,punktmannigfaltigkeiten4
 ,bericht1 erstattet der deutschen      
,mathematiker-,vereinigung1 ,leipzig1   
#aijh4  #ajj ,ibid41 p4 #g4 ,the        
battle-cry is3 ,gegen jede ,resignation1
 aber auch gegen jede ,scholastik6  #aja
 ,ueber die ,stellung der ,definition in

 der ,axiomatik1 ,jahresber1 der        
deutsch4 ,math4-,ver41 vol4 xx41 #aiaa1 
pp4 #bbb-e4 ,the battle-cry is on p4    
#bef and is3 ,fuer den ,cantorismus aber
 gegen den ,russellismus6  #ajb ,ibid41 
p4 #bea4 ,es ist also1 he exclaims with 
the eloquence of emotion and the emotion
 of eloquence1 nicht die                
,geringschaetzung der ,philosophie1 die 
mich dabei treibt1 sondern die ,liebe   
zur ,mathematik2444  #ajc ,ueber die    
,stellung der ,definition in der        
,axiomatik1 ,jahresber1 der deutsch4    
,math4-,ver41 vol4 xx41 #aiaa1 p4 #bea4 
#ajd ,cf4 for this1 ,m41 vol4 xxii41    
,january #aiab1 pp4 #adi-eh4--;,,ed4    
#aje ,bericht1 #aijh1 p4 #gf1 note2 cf4 
p4 #gb4  #ajf ,e4g4 in ,f4 ,hausdorff's 
review of ,russell's ,principles of     
#aijc in the ,vierteljahrsschr4 fuer    
wiss4 ,philos4 und ,soziologie4  #ajg   
,cf4 ,m41 vol4 xxv41 #aiae1 pp4         
#ccc-h4--;,,ed4  #ajh ,russell1 ,a4 ,j4 
,m41 vol4 xxx41 #aijh1 p4 #bbf4  #aji   
,loc4 cit41 p4 #bbb4  #aaj ,see         

,appendix ,q4  #aaa ,see ,appendix ,r4  
#aab ,see ,appendix ,s4     ,,chapter   
,,xxxviii  ,,modern ,,logic ,,and ,,some
 ,,philosophical ,,arguments   ,the most
 noteworthy reformation of recent years 
in logic is the discovery and           
development by ,mr4 ,bertrand ,russell  
of the fact that the paradoxes--of      
,burali-,forti1 ,russell1 ,koenig1      
,richard1 and others--which have        
appeared of late years in the           
mathematical theory of aggregates and   
have just been referred to1 are of an   
entirely logical nature1 and that their 
avoidance requires us to take account of
 a principle which has been hitherto    
unrecognized1 and which renders invalid 
several well-known arguments in         
refutation of scepticism1 agnosticism1  
and the statement of a man that he      
asserts nothing4  ,dr4 ,whitehead and   
,mr4 ,russell say3#aac ,the principle   
which enables us to avoid illegitimate  
totalities may be stated as follows3    
',whatever involves all of a collection 

must not be one of the collection1' or  
conversely3 ',if1 provided a certain    
collection had a total1 it would have   
members only definable in terms of that 
total1 then the said collection has no  
total4' ,we shall call this the         
'vicious-circle principle1' because it  
enables us to avoid the vicious circles 
involved in the assumption of           
illegitimate totalities4 ,arguments     
which are condemned by the              
vicious-circle principle will be called 
'vicious-circle fallacies4' ,such       
arguments1 in certain circumstances1 may
 lead to contradictions1 but it often   
happens that the conclusions to which   
they lead are in fact true1 though the  
arguments are fallacious4 ,take1 for    
example1 the law of excluded middle in  
the form 'all propositions are true or  
false4' ,if from this law we argue that1
 because the law of excluded middle is a
 proposition1 therefore the law of      
excluded middle is true or false1 we    
incur a vicious-circle fallacy4 ',all   

propositions' must be in some way       
limited before it becomes a legitimate  
totality1 and any limitation which makes
 it legitimate must make any statement  
about the totality fall outside the     
totality4 ,similarly the imaginary      
sceptic who asserts that he knows       
nothing and is refuted by being asked if
 he knows that he knows nothing1 has    
asserted nonsense1 and has been         
fallaciously refuted by an argument     
which involves a vicious-circle fallacy4
 ,in order that the sceptic's assertion 
may become significant it is necessary  
to place some limitation upon the things
 of which he is asserting his ignorance2
 the proposition that he is ignorant of 
every member of this collection must not
 itself be one of the collection4 ,hence
 any significant scepticism is not open 
to the above form of refutation4  ,in   
fact1 the world of things falls into    
various sets of things of the same type4
 ,for every propositional function      
,greek3 ph7x7 there is a range of values

 of x for which ,greek3 ph7x7 has a     
signification as a true or a false      
proposition4 ,until this theory was     
brought forward1 there were occasionally
 discussions as to whether an object    
which did not belong to the range of a  
certain propositional function possessed
 the corresponding property or not4     
,thus1 ,jevons1 in early days1#aad was  
of opinion that virtue is neither black 
nor not-black because it is not         
coloured1 but rather later#aae he       
admitted that virtue is not             
triangular4#aaf  ,,footnotes3  #aac ,pa4
 ,ma41 p4 #dj4  #aad ,s4 o4 ,s4 pp4     
#cf-g4  #aae ,e4 ,l4 ,l41 pp4 #abj-a4   
#aaf ,it may perhaps be added that1 some
 years after ,mr4 ,r99ss99ll's death1   
,dr4 ,whitehead stated1 in an address   
delivered in #aiaf and reprinted in his 
book on ,the ,organisation of ,thought  
7,london1 #aiag1 p4 #abj71 that the     
specific heat of virtue is #j4jjc is1 ,i
 should imagine1 not a proposition at   
all1 so that it is neither true nor     

false4444--,,ed4     ,,chapter ,,xxxix  
,,the ,,hierarchy ,,of ,,jokes   ,jokes 
may be divided into various types4 ,thus
 a joke or class of jokes can only be   
the subject of a joke of higher order4  
,otherwise we would get the same        
vicious-circle fallacy which gives rise 
to so many paradoxes in logic and       
mathematics4 ,a certain ,oxford scholar 
succeeded1 to his own satisfaction1 in  
reducing all jokes to primitive types1  
consisting of thirty-seven proto-,aryan 
jokes4 ,when any proposition was        
propounded to him1 he would reflect and 
afterwards pronounce on the question as 
to whether the proposition was a joke or
 not4 ,if he decided1 by his theory1    
that it was a joke1 he would solemnly   
say3 ,there is that joke4 ,if this      
narration is accepted as a joke1 since  
it cannot be reduced to one of the      
proto-,aryan jokes under pain of leading
 us to commit a vicious-circle fallacy1 
we must conclude that there is at least 
one joke which is not proto-,aryan2 and1

 in fact1 is of a higher type4 ,there is
 no great difficulty in forming a       
hierarchy of jokes of various types4    
,thus a joke of the fourth type 7or     
order7 is as follows3 ,a joke of the    
first order was told to a ,scotchman1   
who1 as we would expect1 was unable to  
see it4#aag ,the person 7,a7 who told   
this joke told the story of how the joke
 was received to another ,scotchman     
thereby making a joke about a joke of   
the first order1 and thus making a joke 
of the second order4 ,a remarked on this
 joke that no joke could penetrate the  
head of the ,scotchman to whom the joke 
of the first order was told1 even if it 
were fired into his head with a gun4    
,the ,scotchman1 after severe thought1  
replied3 ,but ye couldn't do that1 ye   
know6 ,a repeated the whole story1 which
 constituted a joke of the third order1 
to a third ,scotchman4 ,this last       
,scotchman again1 after prolonged       
thought1 replied3 ,he had ye there6     
,this whole story is a joke of the      

fourth order4  ,most known jokes are of 
the first order1 for the simple reason  
that the majority of people find that   
the slightest mental effort effectually 
destroys any perception of humour4 ,it  
seems to me that a joke becomes more    
pleasurable in proportion as logical    
faculties are brought into play by it2  
and hence that logical power is allied1 
or possibly identical1 with the power of
 grasping more subtle jokes4 ,the jokes 
which amuse the frequenters of          
music-halls1 ,conservatives1 and ,m4    
,bergson--and which usually deal with   
accidents1 physical defects1            
mothers-in-law1 foreigners1 or over-ripe
 cheese--are usually jokes of the first 
order4 ,jokes of the second1 and even of
 the third1 order appeal to ordinary    
well-educated people2 jokes of higher   
order require either special ability or 
a sound logical training on the part of 
the hearer if the joke is to be         
appreciated2 while jokes of transfinite 
order presumably only excite the        

inaudible laughter of the gods4         
,,footnotes3  #aag ,it may be that1 like
 certain remarks about cheese and       
mothers-in-law 7see below71 the         
statement that ,scotchmen cannot see    
jokes is a joke of the first            
order4--,,ed4     ,,chapter ,,xl  ,,the 
,,evidence ,,of ,,geometrical           
,,propositions   ,it has often been     
maintained that the twentieth           
proposition of the first book of        
,euclid--that two sides of a triangle   
are together greater than the third     
side--is evident even to asses4 ,this   
does not1 however1 seem to me generally 
true4 ,i once asked a coastguardsman the
 distance from ,a to ,b2 he replied3    
,eight miles4 ,on further inquiry ,i    
elicited the fact that the distance from
 ,a to ,c was two miles and the distance
 from ,c to ,b was twenty-two miles4    
,now the paths from ,a to ,b and from ,c
 to ,b were by sea2 while the path from 
,a to ,c was by land4 ,hence if the path
 by land was rugged and the distance    

along the road was two miles1 it would  
appear that the coastguardsman believed 
that not only could one side of a       
triangle be greater than the other two1 
but that one straight side of a triangle
 might be greater than one straight side
 and any curvilinear side of the same   
triangle4 ,the only escape from part of 
this astonishing creed would be by      
assuming that the distance of two miles 
from ,a to ,c was measured as the crow  
flies1 while the road ,a to ,c was so   
hilly that a pedestrian would traverse  
more than fourteen miles when proceeding
 from ,a to ,c4 ,then indeed the        
coastguardsman could maintain the true  
proposition that there is at least one  
triangle ,,abc1 with the side ,,ac      
curvilinear1 such that the sum of the   
lengths of ,,ab and ,,ac is greater than
 the length of ,,bc1 and only deny the  
twentieth proposition of the first book 
of ,euclid4  ,reasoning with the        
coastguardsman only had the effect of   
his adducing the authority of one       

,captain ,jones in support of the       
accuracy of his data4 ,possibly ,captain
 ,jones held strange views as to the    
influence of temperature or other       
physical circumstances1 or even the     
nature of space itself1 on the lengths  
of lines in the neighbourhood of the    
triangle ,,abc4     ,,chapter ,,xli     
,,absolute ,,and ,,relative ,,position  
,some people maintain that position in  
space or time must be relative because1 
if we try to determine the position of a
 body ,a1 if bodies ,b1 ,c1 ,d with     
respect to which the position of ,a     
could be determined were not present1 we
 should be trying to determine something
 about ,a without having our senses     
affected by other things4 ,these people 
seem to me to be like the cautious guest
 who refused to say anything about his  
host's port-wine until he had tasted red
 ink4  ,wherein1 then1 says ,mr4        
,russell1#aah lies the plausibility of  
the notion that all points are exactly  
alike8 ,this notion is1 ,i believe1 a   

psychological illusion1 due to the fact 
that we cannot remember a point so as to
 know it when we meet again4 ,among     
simultaneously presented points it is   
easy to distinguish2 but though we are  
perpetually moving1 and thus being      
brought among new points1 we are quite  
unable to detect this fact by our       
senses1 and we recognize places only by 
the objects they contain4 ,but this     
seems to be a mere blindness on our     
part--there is no difficulty1 so far as 
,i can see1 in supposing an immediate   
difference between points1 as between   
colours1 but a difference which our     
senses are not constructed to be aware  
of4 ,let us take an analogy3 ,suppose a 
man with a very bad memory for faces2 he
 would be able to know1 at any moment1  
whether he saw one face or many1 but he 
would not be aware whether he had seen  
any of the faces before4 ,thus he might 
be led to define people by the rooms in 
which he saw them1 and to suppose it    
self-contradictory that new people      

should come to his lectures1 or that old
 people should cease to do so4 ,in the  
latter point at least it will be        
admitted by lecturers that he would be  
mistaken4 ,and as with faces1 so with   
points--inability to recognize them must
 be attributed1 not to the absence of   
individuality1 but merely to our        
incapacity4  ,another form of this      
tendency is shown by ,kronecker1 ,borel1
 ,poincare1 and many other              
mathematicians1 who refuse mere logical 
determination of a conception and       
require that it be actually described in
 a finite number of terms4 ,these       
eminent mathematicians were anticipated 
by the empirical philosopher who would  
not pronounce that the law of thought   
that ,a is either in the place ,b or not
 is true until he had looked to make    
sure4 ,this philosopher was of the same 
school as ,j4 ,s4 ,mill and ,buckle1 who
 seem to have maintained implicitly not 
only that1 in view of the fact that the 
breadth of a geometrical line depends   

upon the material out of which it is    
constructed1 or upon which it is drawn1 
that there ought to be a paste-board    
geometry1 a stone geometry1 and so      
on2#aai but also that the foundations of
 logic are inductive in their           
nature4#abj ,we cannot1 says ,mill1#aba 
conceive a round square1 not merely     
because no such object has ever         
presented itself in our experience1 for 
that would not be enough4 ,neither1 for 
anything we know1 are the two ideas in  
themselves incompatible4 ,to conceive a 
body all black and yet white would only 
be to conceive two different sensations 
as produced in us simultaneously by the 
same object--a conception familiar to   
our experience--and we should probably  
be as well able to conceive a round     
square as a hard square1 or a heavy     
square1 if it were not that in our      
uniform experience1 at the instant when 
a thing begins to be round1 it ceases to
 be square1 so that the beginning of the
 one impression is inseparably          

associated with the departure or        
cessation of the other4 ,thus our       
inability to form a conception always   
arises from our being compelled to form 
another contradictory to it4            
,,footnotes3  #aah ,md41 ,n4 ,s41 vol4  
x41 ,july1 #aija1 pp4 #cac-ad4  #aai ,j4
 ,b4 ,stallo1 ,the ,concepts and        
,theories of ,modern ,physics1 #dth ed41
 ,london1 #aijj1 pp4 #bag-bg4  #abj     
,ibid41 pp4 #adj-d4  #aba ,examination  
of the ,philosophy of ,sir ,william     
,hamilton1 vol4 i4 p4 #hh1 ,amer4 ed4   
,,chapter ,,xlii  ,,laughter   ,it      
seemed advisable to give here#abb some  
views on laughter1 most of which were   
also held by ,mr4 ,r99ss99ll1 though no 
written expression of his views has yet 
been found4 ,in a review#abc of ,m4     
,bergson's book on ,laughter1#abd ,mr4  
,russell has remarked3  ,it has long    
been recognized by publishers that      
everybody desires to be a perfect lady  
or gentleman 7as the case may be72 to   
this fact we owe the constant stream of 

etiquette-books4 ,but if there is one   
thing which people desire even more1 it 
is to have a faultless sense of humour4 
,yet so far as ,i know1 there is no book
 called ',jokes without ,tears1 by ,mr4 
,mc,quedy4' ,this extraordinary lacuna  
has now been filled4 ,those to whom     
laughter has hitherto been an           
unintelligible vagary1 in which one must
 join1 though one could never tell when 
it would break out1 need only study ,m4 
,bergson's book to acquire the finest   
flower of ,parisian wit4 ,by observing a
 very simple formula they will know     
infallibly what is funny and what is    
not2 if they sometimes surprise their   
unlearned friends1 they have only to    
mention their authority in order to     
silence doubt4 ',the attitudes1 gestures
 and movements of the human body1' says 
,m4 ,bergson1 'are laughable in exact   
proportion as that body reminds us of a 
mere machine4' ,when an elderly         
gentleman slips on a piece of           
orange-peel and falls1 we laugh1 because

 his body follows the laws of dynamics  
instead of a human purpose4 ,when a man 
falls from a scaffolding and breaks his 
neck on the pavement1 we presumably     
laugh even more1 since the movement is  
even more completely mechanical4 ,when  
the clown makes a bad joke for the first
 time1 we keep our countenance1 but at  
the fifth repetition we smile1 and at   
the tenth we roar with laughter1 because
 we begin to feel him a mere automaton4 
,we laugh at ,moliere's misers1         
misanthropists and hypocrites1 because  
they are mere types mechanically        
dominated by a master impulse4          
,presumably we laugh at ,balzac's       
characters for the same reason2 and     
presumably we never smile at ,falstaff1 
because he is individual throughout4    
,the review concludes with the          
reflection that it would seem to be     
impossible to find any such formula as  
,m4 ,bergson seeks4 ,every formula      
treats what is living as if it were     
mechanical1 and is therefore by his own 

rule a fitting object of laughter4 ,now1
 this undoubtedly true conclusion has   
been obtained1 as is readily seen1 by a 
vicious-circle fallacy which ,mr4       
,r99ss99ll would hardly have            
committed4--,,ed4  ,,footnotes3  #abb   
,from a remark on p4 #dg above1 it is   
evident that ,mr4 ,r99ss99ll intended to
 write some such chapter as this4  #abc 
,the ,professor's ,guide to ,laughter1  
,the ,cambridge ,review1 vol4 xxxii41   
#aiab1 pp4 #aic-d4  #abd ,laughter1 an  
,essay on the ,meaning of the ,comic1   
,english translation by ,c4 ,brereton   
and ,f4 ,rothwell1 ,london1 #aiaa4      
,,chapter ,,xliii  ,,gedankenexperimente
 ,,and ,,evolutionary ,,ethics   ,the   
,gedankenexperimente1 upon which so much
 weight has been laid by ,mach#abe and  
,heymans1#abf had already been          
investigated by the ,white ,queen1#abg  
who1 however1 seems to have perceived   
that the results of such experiments are
 not always logically valid4 ,the       
psychological founding of logic appears 

to be not without analogy with the      
surprising method of advocates of       
evolutionary ethics1 who expect to      
discover what is good by inquiring what 
cannibals have thought good4 ,i         
sometimes feel inclined to apply the    
historical method to the multiplication 
table4 ,i should make a statistical     
inquiry among school-children1 before   
their pristine wisdom had been biassed  
by teachers4 ,i should put down their   
answers as to what #f times #i amounts  
to1 ,i should work out the average of   
their answers to six places of decimals1
 and should then decide that1 at the    
present stage of human development1 this
 average is the value of #f times #i4   
,,footnotes3  #abe ,see1 e4g41 ,e4 u4   
,i41 pp4 #ahc-bjj4  #abf ,g4 u4 ,e41    
vol4 i4  #abg ,see ,appendix ,t4        
,,appendixes   ,a4 ,,logic ,,and ,,the  
,,principle ,,of ,,identity4  ,t4 ,l4   
,g41 p4 #de3 ',contrariwise1 continued  
,tweedledee1 if it was so1 it might be2 
and if it were so1 it would be3 but as  

it isn't1 it ain't4 ,that's logic4      
99       99       99       99       99  
,s4 ,b41 p4 #aei3 ,the ,professor said3 
,the day is the same length as anything 
that is the same length as it4          
99       99       99       99       99  
,s4 ,b41 p4 #afa3 ,bruno observed that1 
when the ,other ,professor lost himself1
 he should shout3 ,he'd be sure to hear 
hisself1 'cause he couldn't be far off4 
,b4 ,,synthesis ,,of ,,contradictories4 
,t4 ,l4 ,g41 p4 #ga3 ',what a beautiful 
belt you've got on6' ,alice suddenly    
remarked4444 ',at least1' she corrected 
herself on second thoughts1 'a beautiful
 cravat1 ,i should have said--no1 a     
belt1 ,i mean--,i beg your pardon6' she 
added in dismay1 for ,humpty-,dumpty    
looked thoroughly offended1 and she     
began to wish she hadn't chosen that    
subject4 ',if only ,i knew1' she thought
 to herself1 'which was neck and which  
was waist6'   ,c4 ,,empirical           
,,philosophers ,,and ,,mathematics4  ,t4
 ,l4 ,g41 p4 #gi3 '444 ,now if you had  

the two eyes on the same side of the    
nose1 for instance--or the mouth at the 
top--that would be some help4'  ',it    
wouldn't look nice1' ,alice objected4   
,but ,humpty-,dumpty only shut his eyes 
and said3 ',wait till you've tried4'    
99       99       99       99       99  
,t4 ,l4 ,g41 p4 #gb3 ',and if you take  
one from three hundred and sixty-five1  
what remains8'  ',three hundred and     
sixty-four1 of course4'  ,humpty-,dumpty
 looked doubtful4 ',i'd rather see that 
done on paper1' he said4   ,d4 ,,nominal
 ,,definition4  ,t4 ,l4 ,g41 p4 #gc3    
',when ,i used a word1' ,humpty-,dumpty 
said in rather a scornful tone1 'it     
means just what ,i choose it to         
mean--neither more nor less4'  ',the    
question is1' said ,alice1 'whether you 
can make words mean different things4'  
',the question is1' said                
,humpty-,dumpty1 'which is to be        
master--that's all4'   ,e4 ,,conformity 
,,of ,a ,,paradoxical ,,logic ,,with    
,,common-,,sense4  ,t4 ,l4 ,g41 p4 #ajj3

    ,but ,i was thinking of a plan   ,to
 dye one's whiskers green1   ,and always
 use so large a fan   ,that they could  
not be seen4                            
7,verse from ,white ,knight's song47    
,f4 ,,idealists ,,and ,,the ,,laws ,,of 
,,logic4  ,t4 ,l4 ,g41 p4 #eb-c3        
,tweedledee exclaimed3 '444 if he the   
,red ,king left off dreaming about you  
,alice1 where do you suppose you'd be8' 
',where ,i am now1 of course1' said     
,alice4  ',not you6' ,tweedledee        
retorted contemptuously4 ',you'd be     
nowhere4 ,why1 you're only a sort of    
thing in his dream6'  ',if that there   
,king was to wake1' added ,tweedledum1  
'you'd go out--bang6--just like a       
candle6'  ',i shouldn't6' ,alice        
exclaimed indignantly4 ',besides1 if    
,i'm only a sort of thing in his dream1 
what are you1 ,i should like to know8'  
',ditto1' said ,tweedledum4442 'you know
 very well you're not real4'  ',i am    
real6' said ,alice1 and began to cry4   
99       99       99       99       99  

,t4 ,l4 ,g41 p4 #ig3 ',how can you go on
 talking so quietly1 head downwards8'   
,alice asked1 as she dragged him out by 
the feet1 and laid him in a heap on the 
bank4  ,the ,knight looked surprised at 
the question4 ',what does it matter     
where my body happens to be8' he said4  
',my mind goes on working all the same4 
,in fact1 the more head downwards ,i am1
 the more ,i keep inventing new things4'
         99       99       99       99  
99  ,t4 ,l4 ,g41 p4 #ih3 '444 ,everybody
 that hears me sing--either it brings   
the tears into their eyes1 or else----' 
',or else what8' said ,alice1 for the   
,knight had made a sudden pause4  ',or  
else it doesn't1 you know4'   ,g4       
,,distinction ,,between ,,sign ,,and    
,,signification4  ,t4 ,l4 ,g41 pp4      
#ih-i3 ',the name of the song is called 
,haddocks' ,eyes4'  ',oh1 that's the    
name of the song1 is it8' ,alice said1  
trying to feel interested4  ',no1 you   
don't understand1' the ,knight said     
looking a little vexed4 ',that's what   

the name is called4 ,the name really is 
,the ,aged ,aged ,man4'  ',then ,i ought
 to have said ,that's what the song is  
called8' ,alice corrected herself4      
',no1 you oughtn't3 that's another      
thing4 ,the song is called ,ways and    
,means3 but that's only what it's       
called1 you know6'  ',well1 what is the 
song1 then8' said ,alice1 who was by    
this time completely bewildered4  ',i   
was coming to that1' the ,knight said4  
',the song really is ,a-sitting on a    
,gate4444'   ,h4 ,,nominalism4  ,a4 ,a4 
,w41 p4 #gj3 ',then you should say what 
you mean1' the ,march ,hare went on4    
',i do1' ,alice hastily replied2 'at    
least--at least ,i mean what ,i         
say--that's the same thing1 you know4'  
',not the same thing a bit6' said the   
,hatter4 ',why1 you might just as well  
say that ,i see what ,i eat is the same 
thing as ,i eat what ,i see4'  ',you    
might just as well say1' added the      
,march ,hare1 'that ,i like what ,i get 
is the same thing as ,i get what ,i     

like6'  ',you might just as well say1'  
added the ,dormouse1 which seemed to be 
talking in its sleep1 'that ,i breathe  
when ,i sleep is the same as ,i sleep   
when ,i breathe6'  ',it is the same     
thing with you1' said the ,hatter2 and  
here the conversation dropped1444   ,i4 
,,utility ,,of ,,symbolic ,,logic4  ,a4 
,a4 ,w41 p4 #ib3 ',i quite agree with   
you1' said the ,duchess1 'and the moral 
of that is--,be what you would seem to  
be--or if you'd like it put more        
simply--,never imagine yourself not to  
be otherwise than what it might appear  
to others that what you were or might   
have been was not otherwise than what   
you had been would have appeared to them
 to be otherwise4'  ',i think ,i should 
understand that better1' ,alice said    
very politely1 'if ,i had it written    
down3 but ,i can't quite follow it as   
you say it4'  ',that's nothing to what  
,i could say if ,i chose1' the ,duchess 
replied1 in a pleased tone4   ,j4       
,,mistake ,,as ,,to ,,the ,,nature ,,of 

,,criticism4  ,t4 ,l4 ,g41 p4 #aje3     
',she's in that state of mind1' said the
 ,white ,queen1 'that she wants to deny 
something--only she doesn't know what to
 deny4'  ',a nasty1 vicious temper1' the
 ,white ,queen remarked2 and then there 
was an uncomfortable silence for a      
minute or two4   ,k4 ,a ,,criterion ,,of
 ,,truth4  ,h4 ,s41 p4 #c3    ,just the 
place for a ,snark6 ,i have said it     
twice3   ,that alone should encourage   
the crew4   ,just the place for a       
,snark6 ,i have said it thrice3   ,what 
,i tell you three times is true4        
99       99       99       99       99  
,h4 ,s41 p4 #ej3    ',tis the note of   
the ,jubjub6 ,keep count4 ,i entreat2   
,you will find ,i have told it you      
twice4   ',tis the song of the ,jubjub6 
,the proof is complete1   ,if only ,i've
 stated it thrice4   ,l4 ,,universal    
,,and ,,particular ,,propositions4  ,t4 
,l4 ,g41 p4 #dj3 ,the ,gnat had told    
,alice that the ,bread-and-butterfly    
lives on weak tea with cream in it2 so3 

',supposing it couldn't find any8' she  
suggested4  ',then it would die1 of     
course4'  ',but that must happen very   
often1' ,alice remarked thoughtfully4   
',it always happens1' said the ,gnat4   
,m4 ,,denoting4  ,t4 ,l4 ,g41 p4 #dc3   
,tweedledum and ,tweedledee were1 in    
many respects1 indistinguishable1 and   
,alice1 walking along the road1 noticed 
that whenever the road divided there    
were sure to be two finger-posts        
pointing the same way1 one marked ',,to 
,,tweedledum',s ,,house' and the other  
',,to ,,the ,,house ,,of ,,tweedledee4' 
',i do believe1' said ,alice at last1   
'that they live in the same house6444'  
,n4 ,,non-,,entity4  ,t4 ,l4 ,g41 p4    
#hg3 ',i always thought they human      
children were fabulous monsters6' said  
the ,unicorn4444  ',do you know said    
,alice1 ,i always thought ,unicorns were
 fabulous monsters1 too6 ,i never saw   
one alive before6'  ',well1 now that we 
have seen each other1' said the         
,unicorn1 'if you'll believe in me1     

,i'll believe in you4 ,is that a        
bargain8'         99       99       99  
99       99  ,t4 ,l4 ,g41 pp4 #hj-a3 ',i
 see nobody on the road1' said ,alice4  
',i only wish ,i had such eyes1' the    
,white ,king remarked in a fretful tone4
 ',to be able to see ,nobody6 ,and at   
that distance1 too6 ,why1 it's as much  
as ,i can do to see real people by this 
light6'         99       99       99    
99       99  ,a4 ,a4 ,w41 p4 #ag3 ,and  
she ,alice tried to fancy what the flame
 of a candle looks like after the candle
 is blown out1 for she could not        
remember ever having seen such a thing4 
99       99       99       99       99  
,a4 ,a4 ,w41 p4 #fh3 444 ,this time it  
the ,cheshire ,cat vanished quite       
slowly1 beginning with the end of the   
tail1 and ending with the grin1 which   
remained some time after the rest of it 
had gone4  ',well6 ,i've often seen a   
cat without a grin1' thought ,alice2    
'but a grin without a cat6 ,it's the    
most curious thing ,i ever saw in all my

 life6'         99       99       99    
99       99  ,a4 ,a4 ,w41 p4 #gg3 444   
,the ,dormouse went on14442 'and they   
drew all manner of things--everything   
that begins with an ,m4'  ',why with an 
,m8' said ,alice4  ',why not8' said the 
,march ,hare4  ,alice was silent4  444  
,the ,dormouse went on3 '--that begins  
with an ,m1 such as mouse-traps1 and the
 moon1 and memory1 and muchness1 you    
know you say things are much of a       
muchness--did you ever see such a thing 
as a drawing of a muchness8'  ',really1 
now you ask me1' said ,alice1 very much 
confused1 ',i don't think----'  ',then  
you shouldn't talk1' said the ,hatter4  
,o4 ,,objects ,,of ,,mathematical       
,,logic4  ,t4 ,l4 ,g41 p4 #ic3 ',i was  
wondering what the mouse-trap fastened  
to the ,white ,knight's saddle was for1'
 said ,alice4 ',it isn't very likely    
there would be any mice on the horse's  
back4'  ',not very likely1 perhaps1'    
said the ,knight1 'but1 if they do come1
 ,i don't choose to have them running   

all about4'  ',you see1' he went on     
after a pause1 'it's as well to be      
provided for everything4 ,that's the    
reason the horse has all these anklets  
round his feet4'  ',but what are they   
for8' ,alice asked in a tone of great   
curiosity4  ',to guard against the bites
 of sharks1' the ,knight replied4   ,p4 
,,the ,,principle ,,of ,,permanence4    
,t4 ,l4 ,g41 p4 #ajf3 ',can you do      
,subtraction8 said the ,red ,queen ,take
 nine from eight4'  ',nine from eight ,i
 can't1 you know1' ,alice replied very  
readily 'but--'  ',she can't do         
,substraction1' said the ,white ,queen4 
99       99       99       99       99  
,a4 ,a4 ,w41 p4 #ef3 ,said the ,pigeon  
to ,alice3 '444 ,no1 no6 ,you're a      
serpent2 and there's no use denying it4 
,i suppose you'll be telling me next    
that you never tasted an egg6'  ',i have
 tasted eggs certainly1' said ,alice1   
who was a very truthful child2 'but     
little girls eat eggs quite as much as  
serpents do1 you know4'  ',i don't      

believe it1' said the ,pigeon2 'but if  
they do1 why then they're a kind of     
serpent1 that's all ,i can say4'  ,this 
was such a new idea to ,alice1 that she 
was quite silent for a minute or two1   
which gave the ,pigeon the opportunity  
of adding1 ',you're looking for eggs1 ,i
 know that well enough2 and what does it
 matter to me whether you're a little   
girl or a serpent8'  ',it matters a good
 deal to me1' said ,alice hastily2444   
99       99       99       99       99  
,a4 ,a4 ,w41 p4 #ge3 ',but why asked    
,alice did they live at the bottom of a 
well8'  ',take some more tea1' the      
,march ,hare said to ,alice1 very       
earnestly4  ',i've had nothing yet1'    
,alice replied in an offended tone1 'so 
,i can't take more4'  ',you mean you    
can't take less1' said the ,hatter3     
'it's very easy to take more than       
nothing4'   ,q4 ,,mathematicians'       
,,treatment ,,of ,,logic4  ,a4 ,a4 ,w41 
p4 #gd3 ,the ,hatter had told of his    
quarrel with ,time1 and of ,time's      

refusal now to do anything he asked3    
'444 ,it's always six o'clock now6'  ,a 
bright idea came into ,alice's head4    
',is that the reason so many tea things 
are put out here8' she asked4  ',yes1   
that's it1' said the ,hatter1 with a    
sigh3 'it's always tea time1 and we've  
no time to wash the things between      
whiles4'  ',then you keep moving round1 
,i suppose8' said ,alice4  ',exactly    
so1' said the ,hatter3 'as the things   
get used up4'  ',but what happens when  
you come to the beginning again8' ,alice
 ventured to ask4  ',suppose we change  
the subject1' the ,march ,hare          
interrupted1 yawning4 ',i'm getting     
tired of this4'         99       99     
99       99       99  ,a4 ,a4 ,w41 p4   
#ii3 ',and how many hours a day did you 
do lessons8' said ,alice1 in a hurry to 
change the subject4  ',ten hours the    
first day1' said the ,mock ,turtle1     
'nine the next1 and so on4'  ',what a   
curious plan6' exclaimed ,alice4        
',that's the reason they're called      

lessons1' the ,gryphon remarked1        
'because they lessen from day to day4'  
,this was quite a new idea to ,alice1   
and she thought it over a little before 
she made her next remark4 ',then the    
eleventh day must have been a holiday4' 
',of course it was1' said the ,mock     
,turtle4  ',and how did you manage on   
the twelfth8' ,alice went on eagerly4   
',that's enough about lessons1' the     
,gryphon interrupted in a very decided  
tone4444   ,r4 ,,method ,,in            
,,mathematics ,,and ,,logic4  ,a4 ,a4   
,w41 p4 #ga3 ',two days wrong6' sighed  
the ,hatter4 ',i told you butter        
wouldn't suit the works6' he added1     
looking angrily at the ,march ,hare4    
',it was the best butter1' the ,march   
,hare meekly replied4  ',yes1 but some  
crumbs must have got in as well1' the   
,hatter grumbled2 'you shouldn't have   
put it in with the bread-knife4'  ,the  
,march ,hare took the watch and looked  
at it gloomily3 then he dipped it into  
his cup of tea1 and looked at it again3 

but he could think of nothing better to 
say than his first remark1 ',it was the 
best butter1 you know4'   ,s4 ,,verdict 
,,that ,,logic ,,is ,,philosophy4  ,a4  
,a4 ,w41 pp4 #aai-bc3 444 ',consider    
your verdict1' he the ,king said to the 
jury1 in a low trembling voice4         
',there's more evidence to come yet1    
please your ,majesty1' said the ,white  
,rabbit1 jumping up in a great hurry3   
'this paper has just been picked up4'   
',what's in it8' said the ,queen4  ',i  
haven't opened it yet1' said the ,white 
,rabbit1 'but it seems to be a letter   
written by the prisoner to--to          
somebody4'  ',it must have been that1'  
said the ,king1 'unless it was written  
to nobody1 which isn't usual1 you know4'
  ',who is it directed to8' said one of 
the jurymen4  ',it isn't directed at    
all1' said the ,white ,rabbit1 'in fact 
there's nothing written on the outside4'
 ,he unfolded the paper as he spoke1 and
 added1 ',it isn't a letter1 after all3 
it's a set of verses4'  ',are they in   

the prisoner's handwriting8' asked      
another of the jurymen4  ',no they're   
not1' said the ,white ,rabbit1 'and     
that's the queerest thing about it4'    
7,the jury all looked puzzled74  ',he   
must have imitated somebody else's      
hand1' said the ,king4 7,the jury       
brightened up again47  ',please your    
,majesty1' said the ,knave1 ',i didn't  
write it1 and they can't prove that ,i  
did3 there's no name signed at the end4'
  ',if you didn't sign it1 said the     
,king1 that only makes the matter worse4
 ,you must have meant some mischief1 or 
else you'd have signed your name like an
 honest man4'  ,there was a general     
clapping of hands at this3 it was the   
first really clever thing the ,king had 
said that day4  ',that proves his guilt1
 of course1' said the ,queen1 'so1 off  
with----'  ',it doesn't prove anything  
of the sort6' said ,alice4 ',why1 you   
don't even know what they're about6'    
',read them1' said the ,king4  ,the     
,white ,rabbit put on his spectacles4   

',where shall ,i begin1 please your     
,majesty8' he asked4  ',begin at the    
beginning1' the ,king said very gravely1
 'and go on till you come to the end3   
then stop4'  ,there was dead silence in 
the court1 whilst the ,white ,rabbit    
read out these verses3    ',they told me
 you had been to her1   ,and mentioned  
me to him2   ,she gave me a good        
character1   ,but said ,i could not     
swim4    ,he sent them word ,i had not  
gone   7,we know it to be true73   ,if  
she should push the matter on1   ,what  
would become of you8    ,i gave her one1
 they gave him two1   ,you gave us three
 or more2   ,they all returned from him 
to you1   ,though they were mine before4
    ,if ,i or she should chance to be   
,involved in this affair1   ,he trusts  
to you to set them free   ,exactly as   
they were4    ,my notion was that you   
had been   7,before she had this fit7   
,an obstacle that came between   ,him1  
and ourselves1 and it4    ,don't let him
 know she liked them best1   ,for this  

must ever be   ,a secret kept from all  
the rest1   ,between yourself and me4'  
',that's the most important piece of    
evidence we've heard yet1' said the     
,king1 rubbing his hands1 'so now let   
the jury----'  ',if any one of them can 
explain it1' said ,alice 7she had grown 
so large in the last few minutes that   
she wasn't a bit afraid of interrupting 
him71 ',i'll give him sixpence4 ,i don't
 believe there's an atom of meaning in  
it4'  ,the jury all wrote down on their 
slates1 ',she doesn't believe there's an
 atom of meaning in it1' but none of    
them attempted to explain the paper4    
',if there's no meaning in it1' said the
 ,king1 'that saves a world of trouble1 
you know1 as we needn't try to find any4
 ,and yet ,i don't know1' he went on1   
spreading out the verses on his knee and
 looking at them with one eye2 ',i seem 
to see some meaning in them after all4  
-- said ,i could not swim2 you can't    
swim1 can you8' he added1 turning to the
 ,knave4  ,the ,knave shook his head    

sadly4 ',do ,i look like it8' he said4  
7,which he certainly did not1 being made
 entirely of cardboard47  ',all right1  
so far1' said the ,king2 and he went on 
muttering over the verses to himself3   
'',we know it to be true'--that's the   
jury1 of course--',if she should push   
the matter on'--that must be the        
,queen--',what would become of you8'    
,what indeed6--',i gave her one1 they   
gave him two6' why1 that must be what he
 did with the tarts1 you know----'      
',but it goes on1 ',they all returned   
from him to you1'' said ,alice4  ',why1 
there they are6' said the ,king1        
triumphantly pointing to the tarts on   
the table4 ',nothing can be clearer than
 that4 ,then again--',before she had    
this fit'--you never had fits1 my dear1 
,i think8' he said to the ,queen4       
',never6' said the ,queen furiously1    
throwing an inkstand at the ,lizard as  
she spoke4 7,the unfortunate little     
,bill had left off writing on his slate 
with one finger1 as he found it made no 

mark2 but he now hastily began again1   
using the ink that was trickling down   
his face1 as long as it lasted47  ',then
 the words don't fit you1' said the     
,king1 looking round the court with a   
smile4 ,there was a dead silence4       
',it's a pun6' the ,king added in an    
angry tone1 and everybody laughed4      
',let the jury consider their verdict1' 
the ,king said1 for about the twentieth 
time that day4  ',no1 no6' said the     
,queen4 ',sentence first--verdict       
afterwards4'  ',stuff and nonsense6'    
said ,alice loudly4 ',the idea of having
 the sentence first6'  ',hold your      
tongue6' said the ,queen1 turning       
purple4444   ,t4 ,,gedankenexperimente4 
,t4 ,l4 ,g41 p4 #fa3 ,alice laughed4    
',there's no use trying1' she said3 'one
 can't believe impossible things4'  ',i 
daresay you haven't had much practice1' 
said the ,white ,queen4 ',when ,i was   
your age1 ,i always did it for          
half-an-hour a day4 ,why1 sometimes     
,i've believed as many as six impossible

 things before breakfast4'   ,printed in
 ,great ,britain by  ,,unwin ,,brothers1
 ,,limited1 ,,the ,,gresham ,,press1    
,,woking ,,and ,,london