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Full text of "A new law of thought and its logical bearings;"










No. 4. 







Berlin : A. ASHER AND CO. 

Unpjtfi: F. A. BROCKHAUS 

$eto $ork: G. P. PUTNAM S SONS 

JSombag anto Calcutta: MACM1LLAN AND CO., LTD. 

All rights reserved 






Author of A Primer of Logic 


"One of the greatest pains of human nature 
is the pain of a new idea." BAGEHOT. 

Cambridge : 

at the University Press 

191 1 

Cambridge : 



of the three fundamental Laws of Thought, which 
are traditionally regarded as the cardinal principles 
of Formal Logic, are concerned with the relation of pro 
positions to each other. According to the Law of 
Contradiction, two propositions of the form " A is B " 
and " A is not B" cannot both be true. According to 
the Law of Excluded Middle, they cannot both be false. 
Now it is clear that if there is another principle which 
expresses the fundamental condition of the possibility of 
any proposition taken by itself, without reference to 
others, this also must be regarded as a fundamental Law 
of Thought, and as being logically prior to the Laws of 
Contradiction and Excluded Middle. It is the aim of 
Miss Jones in the following pages to show that there is 
such a Law, and to exhibit in detail its vital importance 
in the treatment of the whole range of topics with which 
Formal Logic deals. This Law of " Significant Assertion " 
is formulated as follows : Every Subject of Predication is 
an identity (of denotation) in diversity (of intension). In 
other words, every affirmative proposition asserts, and 
every negative proposition denies, the union of different 
attributes within the unity of the same thing. In every 
affirmative proposition, the subject-term designates some 
thing as characterised in one way, and the predicate 


designates the same thing as characterised in another 
way. This Law of Significant Assertion is substituted 
by Miss Jones for the traditional Law of Identity, as 
expressed in the formula " A is A." "A is A," if it has 
any significance at all, must, she holds, be taken as an 
attempt to express the essential nature of all predication ; 
but so regarded it is plainly untenable; for to say 
" A is A " is merely to say " A " twice, and not to assert 
anything about "A." There is no proposition, unless 
what is characterised as " A " in the subject-term is also 
characterised as "B" in the predicate-term. 

The service which Miss Jones has rendered to Logic 
in this little volume lies not so much in the mere enun 
ciation of the " Law of Significant Assertion " as in her 
thorough and systematic application of it, so as to clear 
up special logical problems. By way of illustration, I may 
refer to her discussion of the doctrine of " the fourfold 
implication of propositions in Connotation and Denota 
tion," and to her account of immediate inferences, and of 
the syllogism. As regards syllogistic inference, it may be 
worth while to refer to a point which Miss Jones has not 
expressly noticed. The Law of Significant Assertion 
supplies the most direct, simple and general vindication 
of the syllogism against the charge of petitio principii. 
The charge is based on the fact that the conclusion asserts 
of the same thing the same predicate which has already 
been ascribed to it in the major premiss. The straight 
forward reply is, that in the conclusion this predicate is 
brought into connexion with an attribute with which it 
has not been connected in either of the premisses. A 


remarks to B, " That woman in the corner is a scare-crow." 
B replies, "Sir, that woman is my wife." For A, it is a 
startling novelty, and no mere repetition, to discover that 
he has called B s wife a scare-crow. The novelty is plainly 
due to a new synthesis of attributes with the same deno 
tation, the combination of the attribute of being B s wife 
with that of being the woman whom A has just called a 

Miss Jones seems to have made out a good case for 
regarding the Law of Significant Assertion as a funda 
mental Law of Thought. But its claim to be the only 
justifiable rendering of the Law of Identity is not so 
clear. The best writers on Logic tend to interpret this 
law as expressing the immutability of truth. According 
to them, it means that the truth of a proposition is 
unaffected by variation of time, place and circumstances, 
or of the minds which apprehend it. Either this prin 
ciple, or, if the pragmatists be right, its contradictory, 
seems to demand recognition as a fundamental law of 
thought, and it is certainly a principle of Identity. But 
it is of course no substitute for the Law of Significant 
Assertion. The question which of the two is the most 
appropriate interpretation of the cryptic formula " A is A " 
is of quite subordinate interest. 


March, 1911. 



FALLACIES . . .. . . , .68 


MY object in the following brief essay is to propound a 
certain analysis of Categorical Propositions of the forms 
$ is P, S is not P, to show that this is the only general 

it. is nofifilU^ t.n }.rrp-nt, anrl t.r inrlinafci ifa 


(1) p. 9 last line, read (2) the intensional, connotational, or 
implicational ; 

(2) p. 20 top, read In A = B there is between A and B equality 
of quantity or value in intensional diversity thus there is not only 
denotational identity between A and equal to B, but also qualitative 
sameness in qualitative diversity between A and B. 

(3) p. 23, line 3 from foot. For E is F read A is E. 

(4) p. 54, lines 16 and 17, delete and these may conflict. 

(5) p. 66, line 6 from foot, delete three. 

(6) p. 71, line 15. For name read names. 

TCM . A is related to B implies A is 





MY object in the following brief essay is to propound a 
certain analysis of Categorical Propositions of the forms 
8 is P, S is not P, to show that this is the only general 
analysis which it is possible to accept, and to indicate its 
bearing upon logical science. According to the analysis 
in question, S is P asserts Identity of Denotation in 
Diversity of Intension, and S is not P denies this. The 
example given by Professor Frege (whose analysis of 8 
is P I understand to agree roughly with mine) is 

" The morning star is the evening star " 

terms " morning star " and " evening star " apply to one 
thing, but the meaning, intension, or qualitative implica 
tion of " morning star " is not the same as that of 
" evening star." " The largest city in the world is the 
Metropolis of England" is another illustration, where 
again it is clear that the two names or terms, the Subject 
and Predicate of the Assertion, apply to one place but 
have different meanings or definitions. S is not P asserts 
Difference of Denotation (Otherness) in Difference of 
Intension (Diversity) e.g. " Cambridge is not Oxford," 

( j . A is related to B implies A is not B, 

00- . / ; 

j. 1 


We need propositions of the form S is P, S is not P, 
for significant assertion, and without them no satisfactory 
statement can be given of the " three fundamental Laws 
of Thought," which are put forward as the basis of logical 
science. The first two of these Laws are commonly 
formulated as: (1) A is A, (2) A is not non-A } and the 
third sometimes as A is either A or non-A (3). Desperate 
efforts have been made by logicians to give a valuable 
meaning to A is A ; but if A is A, interpreted as A is A, 
is retained as the first fundamental Law, there is no 
possible passage from it to A is B, and A is A or A is B 
(S is P) must be given up. This is fully recognised by 
Lotze, who gives up (theoretically) S is P. A is A tells 
us no more than A is A, and if we begin with it, we must 
also end with it, if we are to be consistent. I maintain 
that we must not begin with it, but must begin instead 
with a Law of significant assertion assertion of the forms 
S is P, S is not P, forms which provide the only straight 
forward and effective statement of the second and third 
Laws of Thought, thus : 

S is P [cannot both be true (L. of Contradiction) 
S is not P [cannot both be false (L. of Excluded Middle). 

It follows from these two Laws that of any Subject of 
Predication (S) either P or not-P can be affirmed. Thus 
from them, and S is P, S is not P, analysed as above, we 
obtain the principle that : 

Every Subject of Predication is an Identity-in- 

It follows further that every Predicate (P) is neces 
sarily incompatible with not-P (absence of intension P) 
and necessarily compatible with not-not-P. (This suggests 
a formal principle of necessary connection of attributes.) 


I contend that if we start, not with A is A, but with 
the principle that Every Subject of Predication is an 
identity (of denotation) in diversity (of intension), this 
Law (1), and the Laws of (2) Contradiction and (3) Ex 
cluded Middle (of which (1) for the first time makes 
logically possible the formulation given above) do furnish 
a real and adequate and obvious basis and starting-point 
of " Formal " Logic. Granted propositions of the form 
S is P, with the identity-in-diversity analysis and the 
corresponding analysis of 8 is not P, together with the 
traditional Laws of Contradiction and Excluded Middle, 
the whole scheme of Immediate and Mediate Inference 
can be built up systematically and explicitly, as I hope to 
show. The possibility of Conversion, e.g. implies that the 
Predicate, as well as the Subject, of any Proposition has 
Denotation, and a Denotation that is implicitly quan 
tified; the one indispensable condition of Mediate In 
ference is identity of Denotation of the Middle Term in 
both premisses. Without propositions of the forms S is P, 
8 is not P, thought cannot live or move ; but the 
disastrous acceptance of A is A, with its baffling am 
biguities, has stood in the way of their being rightly 
analysed by logicians and explicitly recognised by them as 
fundamental forms of significant assertion, without which 
not even the Laws of Contradiction and Excluded Middle 
can receive satisfactory expression 1 . 

1 In the following pages I have occasionally borrowed from writings 
of my own in cases where I have not felt able to improve upon the 
statement already printed. 



"I am the pillars of the house, 
The keystone of the arch am I ; 
Take me away, and roof and wall 
Would fall to ruin utterly." 



IT will be admitted that up to the present time no 
adequate and unquestionable basis of the Science of Logic 
has been found that the Method of Logic, itself the 
Science of Method, is not wholly satisfactory. Logic is 
often defined as the Science of the Laws of Thought 
the Laws, that is, of Identity, Contradiction, and Excluded 
Middle ; but on the one hand the statement of these Laws 
is not uniform, and the interpretation of at least the first 
of them, the Law of Identity (A is A, whatever is is, 
Everything is what it is), is matter of perpetual dispute ; 
on the other hand no one of these Laws alone, nor all of 
them together, can or do take account of, or can explain 
and justify, the common indispensable form of Categorical 
Assertion S is P e.g. Trees are green, All Men are 
mortal, George V is the present King of England, Per 
severance is admirable, Honesty is the best policy, The 
quality of Mercy is twice bless d. On the contrary, A is A 
appears to exclude it, and there is no passage from A is A 

to A is B. And if anyone who accepts A is A, and the 
corresponding expression of the Law of Contradiction, A 
is not not- A, is driven into giving A is B or not-B as the 
Law of Excluded Middle, it is for him to show what 
logical connexion there is between the last " Law " and 
the two previous ones. Logic undoubtedly, like all other 
Sciences, like literature, like common thought and common 
speech, uses the forms 8 is P, S is not P uses them at 
every step. It must use them, of course; it has no choice; 
without them, it would be impossible to affirm or deny; 
but it adopts them in the same fashion as Bentham 
adopted the Greatest Happiness of the Greatest number as 
his ultimate ethical principle that is to say, without 
any reasoned justification. No " plain man " certainly, 
would be expected to give any reason why he should use 
propositions of the form A is B rather than of the form 
A is A; lout a, logician who declares that A is A is the 
first Law of Thought, and (if he is consistent), that 
A is not not- A and A is either A or not A 1 are the other 
two, may fairly be called upon to explain the fact that he 
habitually says that Roses are red and Violets are blue, 
rather than Roses are roses, Red is red, Violets are violets, 
and so on. For logicians to find fault with a so-called 
" Law " which is a pure tautology, which is expressed in 
a form which may indeed have important uses, and may 
be employed epigrammatically or rhetorically, but in 
which no ordinary sensible person would think of trying 
to convey straightforward information, or matter of fact 
much less a fundamental principle is no new thing. 

1 This, however, is generally stated A is either B or not B and 
sometimes the A is B form is slipped into even in stating the Law of 
Contradiction, by upholders of the A is A Law of Identity. 

6 A NEW 

To lay it down (1) that we can never legitimately affirm 
of any subject a predicate different from itself, while at 
the same time (2) it has to be allowed that this rule 
cannot be even stated without being broken, without 
using assertions of the form S is P, was we know, a state 
of mind possible in the time of Plato; it was possible 
because those who asserted (1) thought it self-evident 
that the Predicate ought always to be the same as its 
Subject, "that to apply many Predicates to one and the 
same Subject is to make one thing into many things." 
And as for (2), they could not deny it; while to give 
up (1) seemed to be a denial of self-evident truth, to give 
up (2) was sheerly impossible. The situation is rather 

That there is a difficulty about S is P we need not 
question, that logicians who accept A is A are impera 
tively called upon to show how this " Law " can be adapted 
to propositions of form A is B (S is P) is too obvious 
to need pointing out. Some writers have tried to give a 
meaning to A is A which does not seem to prohibit 
diversity of Predicate from Subject a meaning which 
is itself expressed in the A is B, not in the tautological 
A is A, form ; Mr Bradley e.g. interprets the Law of 
Identity to mean that "if what I say is really true, it 
stands for ever." A is A thus expounded into A is B 
does not of course exclude propositions of A is B form. 
Dr Bosanquet frankly admits that, while he would not 
accept either A is B or A is A as a schematic ex 
pression of the Law of Identity, he would prefer A is B 
to A is A 1 . 

1 " If I were asked " he says, " how I should represent a true 
Identity, such as a judgment must express, in a schematic form with 


The only logician, as far as I know, who, while re 
taining A is A in its purity has made a determined effort 
to reconcile it with propositions of the A is B (8 is P) 
form, is Lotze. He holds (Logic, Bk I. ch. n.) that "our 
thought is subject to a limitation, has to conform to a 
law... in the categorical judgment each constituent can 
only be conceived as self-same [= ?]. This primary law of 
thought, the principle of identity, we express positively 
in the formula A = A." He states the conclusion to 

symbolic letters, I should say the problem was insoluble. Every A is B 
would be much better than Every A is A; but as the letters are not parts 
in any whole of meaning, they are things cut asunder with an axe ." 
(Dr Bosanquet in Mind, 1888, p. 357.) (The objection that in A is B 
"the letters are not parts in any whole of meaning" seems either 

(1) inaccurate, for there is a symbolic whole, viz. (A, B, I which has 

a meaning and an important one, or (2) irrelevant, if what is meant is 
a concrete special " whole of meaning.") 

It is clear from other passages in the same article that for 
Dr Bosanquet, individual identity is not distinguished from qualitative 
one-ness of two things e.g., he speaks of some "present impression" 
as being identical with a former impression " (p. 360), and says that 
" the element of identity between two outlines can be accurately pointed 
out and limited, but the moment they cease to be two, it ceases to be an 
identity" (p. 359). He objects to drawing "a sharp line between the 
unity of the individual human being... and the unity of human beings 
in identical sentiments, ideas, purposes or habits" (p. 362), and says 
that a number of persons may have "a really identical purpose and 
endeavour and consciousness of certain facts" (p. 364). Again (p. 365), 
he says "Any indiscernible resemblance [ = ?] between two different 
contents, in specified respects, will do whatever identity will do, because 
it is identity under another name" (if so, what need is there of a 
Distributed Middle in Syllogism?); and on p. 366 speaks of "indis 
cernible likeness [ = ?] or identity." With this meaning of identity it 
certainly is not clear how "a true identity" could be satisfactorily 
expressed as A is B. Connotationally, qualitatively, A is not B. 


which he is driven, thus : " This absolute connexion of 
two concepts 8 and P, in which the one is unconditionally 
the other and yet both stand over against each other as 
different, is a relation quite impracticable in thought : 
by means of this copula, the simple is of the cate 
gorical judgment, two different contents cannot be con 
nected at all ; they must either fall entirely within one 
another, or they must remain entirely separate, and the 
impossible judgment ( S is P resolves itself into the 
three others, ( S is 8, P is P, 1 S is not P ." (Engl. 
transl. p. 59.) 

Whether A is A is understood as A-ness is A-ness, or 
in any other possible way in which A is A is honestly 
interpreted as A is A (not as A is B), the acceptance of 
it as a first and fundamental Law is absolutely suicidal 
for Logic from a theoretical point of view. But it must 
be confessed that its nominal acceptance does not appear 
to have seriously affected the construction of the Science. 
A is A cannot justify or support this, it even seems in 
consistent with it, but the restrictions logically imposed 
by A is A have (almost universally) been not only not 
respected, they have not even been borne in mind, and 
A is A itself has received a variety of interpretations 
(generally of the form A is B) which it was natural to 
ignore as they mostly did not interfere with either theory 
or practice, and it was thus easy for logicians to go on 
systematising and constructing in complete independence 
of the " First Law of Thought." 

No doubt the speculative incompatibility between it 
and ordinary assertion has been for the most part a 
"contradiction that was not seen." When it has been 
seen, common sense has had no hesitation in driving a coach- 


and-six through the venerable but insubstantial obstacle. 
Lotze, keenly aware of the contradiction and loyal to 
tradition, but oblivious for the moment of the needs and 
actualities of living thought, imagined that he must, 
and could, give up S is P. The actual starting-point 
of Logic has been not A is A, but the Law of Contra 
diction and the Law of Excluded Middle, and the effort 
to analyse S is P (not-P) , and in Conversion, Mediate 
Inference etc., it is propositions of those forms that have 
been dealt with. But those forms were accepted un 
critically, and together with A is A. Logic has lacked a 
First Law which could furnish a legitimate and logical 
starting-point and be capable of development and general 
application, have a real and important difference from, 
and connexion with, the Law of Contradiction and the 
Law of Excluded Middle, be effective throughout the 
Science of Logic, and justify, explain and support logical 
procedure. Though A is A may be sometimes a con 
venient mode of expression, we cannot start from it as 
the fundamental prepositional form and we do not see 
how to get from it to A is B. A is B is the inevitable 
point of departure, and this has, as the limit on one side 
(the side of tautology) A is A (which excludes diversity 
of intension), and on the other (the side of Contra 
diction), A is not- A (which excludes identity of denota 
tion). A is A, f A, Aj 5 is of course quite different from 

I think that every name or term has two aspects : 

(1) the denotational, extensional, or applicational ; 

(2) the intensional or connotational ; 

10 A NEW 

corresponding to the two aspects of the things of which 
they are names i.e. the aspects of (1) Thatness and 
(2) Whatness, to use Mr Bradley s terms. Everything 
of which we can think or speak is (1) Something and 
(2) some definite sort of something. Everything must be 
thought as having (1) existence (in the widest sense 
mere thing-hood) and (2) some fixed definite nature and 
constitution. For the sake of clearness, I propose in what 
follows to confine the term identity to denotational one 
ness, as distinct from one-ness in the intensional sense, 
which makes possible general names, classing, and classi 
fication. Without both (1) and (2) no assertion is possible, 
nothing can be Subject or Predicate of a proposition, 
The Law of Identity may have been an attempt to 
express the qualitative fixity of nature of anything in 
brief and self-evident form ; if so, the expression A is A 
is unfortunately incapable of expressing what was in 
tended. If it does express a meaning, that meaning is 
clearly not self-evident, for there is nothing about which 
there has been more dispute than the meaning of A is A. 
It seems to me that until we have A is B (S is P) there 
is nothing to accept or reject, nothing to doubt or dispute, 
and that the true significance of contradiction is to deny 
of something some predicate which has already been 
affirmed of it. It might seem that for conceptualists the 
problem of A is A was simplified, as their whole interest 
was in Quality, Intension, as distinct from Extension 
or Thatness; but it is demonstrable that no significant 
affirmation can be purely qualitative. 

If we genuinely accept A is A as the expression of a 
fundamental and primary logical principle, the difficulty 
is, how theoretically to get beyond it. If we reject it, 


what we need, and what we find, to put in its place, is a 
principle of significant Assertion Assertion of the form 
S is P. The laws of Contradiction and Excluded Middle 
are laws of the relations of assertions, and they cannot 
be expressed in satisfactory and unambiguous form with 
out the use of S is P, S is not P, propositions. So even 
for them we require a prior principle, explaining and 
justifying the S is P proposition itself. Such a logical 
principle, based on a new analysis of S is P, I think I can 

I call the analysis in question " new " because although 
I put it forward in print in 1890, and although Dr Keynes 
in his Formal Logic has practically adopted it as appro 
priate to "logical equations" (loc. cit. 4th edit. pp. 189, 
190), it has not received much attention no doubt be 
cause many other accounts of the Categorical Proposition 
have looked so like it (and in fact sometimes came so 
near it) that the fundamental difference has not been 

"Oh, the little more and how much it is, 
And the little less and what worlds away." 

And although my own conviction has remained un 
shaken because the doctrine has seemed to me to stand 
all the tests that I could apply in a thoroughly satisfactory 
manner, I should not have taken up the question again at 
this time but for two circumstances. One is that I have 
rather suddenly become aware that my analysis furnishes 
a law of Categorical Assertion which together with the 
Laws of Contradiction and Excluded Middle stated in 
S is P, S is not P, form does provide Formal Logic with 
an adequate foundation, and gives a systematising prin 
ciple, in complete accord both with common thought and 


common usage, and with the accepted structure of logical 
science, and is perhaps further of direct philosophical im 

The other circumstance to which I refer is, that I have 
recently had my attention drawn to the fact that Professor 
Frege s analysis of Categoricals (published in 1892) was 
apparently the same as my own, and that a similar view 
was adopted by Mr B. Russell (1903) in his Principles 
of Mathematics, where Frege s theory of the import of 
propositions is expounded with sympathetic approbation. 

Recognising in Terms the two aspects of Extension 
(or Denotation) and Intension (as Jevons and most other 
modern writers on Logic do), I approach from that point 
of view the question : How are the propositions of the 
forms 8 is P, S is not P, to be analysed ? I hold that one 
or other of these two symbolic expressions may be applied 
to every Categorical Proposition. Further, that in Pro 
positions of which the Term-names are Class-names e.g. 
All Lions are carnivora conversion, involving Quanti 
fication of the Predicate, is possible and legitimate. 

By the Extension or Denotation of a Term I mean 
the things to which it applies, by its Intension I mean those 
properties or qualities of the things which it signifies. As 
Dr Keynes says : " The extension of a name consists of 
objects of which the name can be predicated ; its in 
tension consists of properties which can be predicated of 
it " (Formal Logic, 4th ed. p. 22). " Intension may be 
used to indicate in the most general way what may be 
called the implicational aspect of names " (loc. cit. p. 26). 
E.g. (1) Quadruped in extension denotes lions, tigers, 
horses, dogs, kittens, etc., etc., in intension it means 
having four feet ; gold in extension applies to this cup, 


that ring, those sovereigns, etc., in intension it means 
yellow, heavy, malleable, insoluble in aqua regia ; man 
in extension denotes Henri Bergson, Josiah Royce, J. J. 
Sargent, Mary Findlater, Jane Barlow, Madame Curie, 
etc., in intension it signifies having rationality and 

It is to be observed that we may know the applica 
tion or extension of a name and not know its intension 
(definition or signification) and vice versa. E.g. I know 
that metal in extension denotes gold, silver, copper, iron, 
lead, tin, mercury, aluminium, etc., and I know these 
when I see them, but I am not able to give a satis 
factory statement of the intension which they have in 

Or again I know, or I may know, all the inhabitants 
of a country parish and be able to greet them correctly 
by name when I meet them, but may be entirely unable 
to give a recognisable description of any of them. Or 
I may know real diamonds from paste, or one disease 
from another, and always apply the names rightly, and yet 
be unable to set out even to myself the connotation or 

On the other hand I may have full descriptive know 
ledge of a person or plant or precious stone, and yet not be 
able to recognise the person or plant or jewel though it may 
much concern me to do so. I may even know much more 
about a person than his ordinary acquaintances, or even 
than his dearest friend, and be able to give a much more 
accurate description of his appearance and manner, and 
yet not know him when I meet him. Or I may recognise, 
though I cannot define, Justice; and define, though I 
might not recognise, a chiliagon. 


Extension and Intension or both may be imaginary. 
I may put together elements universally recognised as 
charming, and draw a fancy portrait or a fictitious character; 
or I may attribute to an actual person impossible or in 
compatible perfections. 

What I insist on is that all the names we use have both 
Extension and Intension ; and either of these may be a 
guide to the other. I may have the things to which 
a name applies put before me (Extensive definition) and 
from examination of them reach the intension : or have 
intension given, and go out and by means of it determine 

It may occasionally be possible and be convenient to 
apply the terms Extension, Intension, to things as well as 
to their names, but I suggest as generally appropriate to 
things and not names, the terms Quiddity and Quality 
for the aspects of Thatness and Whatness in things, and 
Entity for Quiddity + Quality, and for that which is asserted 
in a proposition as distinct from the assertion, I would 
suggest the term Assertum. 

According to my analysis, propositions of the form 
8 is P assert identity of denotation (extension) in diversity 

of intension (s~r) ; while correspondingly S is not P 
asserts difference (or otherness) of denotation in intensional 
diversity (IT) C}T) i. e. it denies identity of denotation. 

In S is not P the intension of P is asserted to be absent 
from what is denoted by S. The purpose of S is not P is 
not to assert that the intension of S is diverse from the 
intension of P that goes without saying, and is essential 
to S is P. The speaker who asserts S is P starts from a 


whole (si?) ; the hearer or learner hears first S, then P, 

and puts the two together into the whole, (s P) , from 

which the speaker started. The is of the S is P can 
not mean sameness of intension (exact similarity) for S 
and P ex vi terminorum are diverse, have different inten 
sions as Lotze avows (cp. ante], two different concepts 
or contents cannot be connected at all by the simple 
" is" of the categorical judgment ; if S and P were taken 
in intension only, we could say of S nothing but that 
S is not P , and if 8 is P did not indicate one-ness of 
denotation, then S and P would not refer to an identical 
object, and we should again have to say S is not P. If 
terms were taken in denotation only, we should not know 
what to do with more than one Term in affirmation. 

An intension S neither (1) excludes from the denota 
tion of S every other intension P, nor (2) does the 
addition of P to S change the intension S. If (1), no 
significant affirmation would be possible ; if (2), we should 
never be sure what we were affirming. The thing which 
is S is of course modified by the addition to it of the 
intension P, but not so the intension S. 

" Suppose I assert that all fronds of the Mountain 
Buckler are erect. The meaning of the assertion is fixed 
and definite, and, if true, it is true once for all. If I 
go on to say that the fronds are also lance-shaped in 
form, pinnately divided, that the pinnae stand opposite 
(generally), that they are narrow and tapering and are 
pinnatifidly divided do any of these affirmations, or all 
of them, in the least alter or modify the meaning of my 
original statement that all the fronds are erect ? It must 


be admitted that no one of them alters the meaning of 
any other; but what is very importantly modified is my 
knowledge of, and power of identifying, the thing itself, 
the actual object in time and space, the Subject of Attri 
butes to which all these successive characteristics are 
assigned. All the successive predicates are related as 
joint characteristics of the whole which they qualify ; they 
are related not as modifying each other, but as modifying it. 

That the interior angles of an isosceles triangle are 
equal to two right-angles, is a general truth, the meaning 
of which is not affected by the further general truth that 
the angles at the base of an isosceles triangle are equal to 
each other, or that any triangle is half of a parallelogram. 
The meaning of the assertion: This is an engraving of 
a picture by Gainsborough, is not modified or changed by 
the further assertions : The picture is a portrait of Lady 
Mulgrave, it is one of the artist s masterpieces, it is 
supposed to be now in America. But though any one of 
these statements does not alter or modify the meaning 
of the others, each one does modify the hearer s knowledge 
concerning the object which is spoken of." (Mind, 1908, 
p. 391, etc.) 

Propositions of forms S is P, S is not P having been 
admitted, and analysed as above (pp. 14, 15), we are of course 
justified in formulating the commonly accepted logical 
Laws of Contradiction and Excluded Middle as follows : 

S is P ) cannot both be true (Law of Contradiction. 
S is not P ) cannot both be false (Law of Excluded Middle). 

Both these Laws appear to be self-evident, and it is 
perhaps partly because A is A has been supposed to be 
self-evident tear e^o^rfv, that it has so long held its 


ground, and that such persevering and desperate efforts 
have been made to give it an interpretation which would 
fit it for the post of the First Law of Thought. If Logic 
were provided with laws really self-evident, laws also on 
which the Science could be based, and according to which 
its structure could be built up, then indeed the founda 
tion and method of Logic would be adequate and satis 
factory. The old Law of Identity, however, affords only 
a simulacrum of self- evidence. The Law of Identity in 
Diversity which I propose to put in its place is simply a 
law of significant assertion a law which while it is arrived 
at through analysis of propositions of the form 8 is P 
(8 is not P) implies conditions which make such assertions 
possible, conditions without which knowledge itself seems 
impossible without assertions of this form, knowledge 
could not be communicated, nor even formulated. I hold 
that there is no alternative and really primary analysis 
which will bear investigation; that all valid interpreta 
tions of categoricals which go beyond this must be based 
upon it. 

That 8 is P asserts an Identity of Extension in 
Diversity of Intension seems to me to be on reflection 
self-evident. I should at any rate claim that its denial 
must be admitted to be inconceivable, and as to the form 
8 is P, as the Eleatic Stranger in the Sophistes observed, 
men " admit it implicitly and involuntarily in their common 
forms of speech, they cannot carry on a conversation with 
out it." 

Taking together the three Laws of Identity in 

Diversity, of Contradiction, and of Excluded Middle, we 

may say that of any Subject (8), P must be affirmed or 

denied, but not both, i.e. of any subject (S), P, or not-P, 

j. 2 


but not both, can be predicated. Thus we reach the 
principle that 

Any Subject of Predication is an identity of denota 
tion in diversity of intension. 

Such subject is a whole to which the two terms S and 
P (or not-P) are applicable as names. For every 8 is P, 

(s p) , or not-P, (snot-PJ ; to the thing spoken of, in both 

cases, diverse intensions, S and P, or S and not-P, are 
assigned, and S and P (or not-P) are equally names of the 

denotational whole (s P) (or ( Snot - p )), and are therefore 

interchangeable, and can be substituted one for the other. 
The Law of Identity just formulated implies, I believe, that 
Everything is an identity of extension in diversity 
of intension. 

At any rate we could only disprove this by showing 
that there is something which is not a Subject of Predica 
tion. But to show this, we must expressly treat it as a 
Subject of Predication. Thus the proof that it is not, 
involves that it is. Everything is a possible Subject of 
Predication, and, directly the question is raised, it becomes 
an actual Subject of Predication. 

In the Law of Contradiction what is asserted is, that 
if the diverse intensions signified by S and P are identical 

in denotation, (SP) , then the diverse intensions S, and 
not-P (absence of intension P), are not identical in de 
notation ( S ) fnot-Pj . What the Law of Excluded 



Middle asserts is that if the intensions signified by S and 
P (or not-P) have not identical denotation, then the in 
tensions S and not-P (or P) have identical denotation. 
P, and not-P (= intension P absent), are mutually exclu 
sive, and together exhaustive of all possibilities. 

Though $ is P is not a self-evident and universally 
applicable proposition, S is P or not-P is both, but we 
should not have been in a position to assert it, unless we 
had first established the forms S is P, S is not P. If it 
were inevitable to analyse S is P as Lotze feels forced to 
do into : 

S is S, 
P is P, 
S is not P, 
no such principle could be formulated. 

>S is P, S is not P, express the two kinds or qualities 
of assertion, affirmative and negative, in the most general 
and abstract way, as a = b may stand for any and every 

There are large and important groups of assertions 
which though they can be expressed as S is P are more 
appropriately exhibited in skeleton form and diagram- 

matically as A is not B, (&j (&j , since what they 

affirm is a relation between two things which have not 
identical extension, however vital the relation between 
them may be. 

E.g., A is equal to B, G is father of D, E is to the 
left of F: 



In A = B there is between A and B equality of 
quantity or value in intensional diversity i.e., there 
is not only denotational identity, but also qualitative 
sameness in qualitative diversity. 

In all the above instances two non-identical things are 
considered in their relation to each other, in the "system" 
of related things (numbers or quantities, family con 
nexions, positions in space) to which they respectively 
belong. But the matter of fact in each case can be 
perfectly well expressed (in any given case) as an identity 
in diversity thus: 

A is equal-to-B, E is to-the-left-of-F, and so on. 

A is not B and E is not F, and it cannot be predicated 
of A that it is B, nor of E that it is F ; but we can predi 
cate of A that it is something equal to B, of E that it is 
something to the left of F. 

The copula in : A is equal-to-B, 

E is to-the-left-of-F 

(as always in S is P propositions) signifies identity of 
denotation, and the special kind of relation between A 
and B, E and F is here expressed in the Predicate. 

In dealing with any " Relative Propositions," a know 
ledge of the special system to which they belong is 
required. The symbolic forms 8 is P, 8 is not P, are the 
only ones that can be applied everywhere, and as they are 
of the extremest generality, they are also of the extremest 
abstractness and simplicity. The Subject and Predicate 

in any 8 is P give the whole (s P) from which it is 
inferrible that P is S, not-P is not-S, not-S is not-P. 
S is not P gives the " system " MM MM ; from which it 


can be inferred that P is not S, not-S is P, not-P is S. Of 
course every variety of systematic relation between S and 
P is possible, as has just been indicated. 

A fortiori arguments are simply a special case of 
arguments which turn upon Relativity of Terms. 

A Proposition which has a relative term for S or P 
or both, besides the ordinary Immediate Inferences (Educ 
tions) which can be drawn from it in just the same way 
as from non-relative Propositions, furnishes other imme 
diate inferences to anyone acquainted with the system to 
which it refers. These inferences cannot be educed except 
by a person knowing the " system " ; on the other hand, 
no knowledge is needed of the objects referred to, 
except a knowledge of their place in the system, and 
this knowledge is in many cases co-extensive with 
ordinary intelligence ; consider, e.g., the relations of mag 
nitude of objects in space, of the successive parts of time, 
of family connexions, of number. From such a Pro 
position as : C is a grandfather of D, in addition to such 
inferences as could be drawn from a non-relative Propo 
sition (a-grandfather-of-D is C, not-a-grandfather-of-D 
is not-C, etc.), it is, of course, possible for anyone having 
the most elementary knowledge of family relationship to 
infer further that : 

D is a grandchild of C, 

A parent of D is a child of C, 

A child of D is a great-grandchild of C, 

The father of C is a great-grandfather of D, etc. 


From C is equal to D (besides Something equal to D is 
C, No not-equal to D is C, etc.), it can be inferred that 

D is equal to C, 

C is not less than D, / (Home thm g) \ (^\ (^\ 

D is not greater than C, \ equal to / \^S V_y 

C is not greater than D, 

Whatever is greater than C is 

greater than D, 
and so on (compare C is an inference from D). 

In each of the above examples we are not dealing with 
one object or group in the same way as in non-relative 
Propositions, e.g., 
All men are 

All men \ / Byzantium 

I I a lark 

Constantinople,^ / \ r 

This bird is a lark ; 
but we are considering, besides the identity of appli 
cation of S and P, two objects denotationally distinct, 
namely C and D. (See my General Logic, p. 34.) 

If, making an advance in complication, and passing to 
Mediate Inference, it is asserted that : 

M is P 
and S is M, 

the hearer is entitled to the construction /s M P) , where 

three diverse intensions are referred to one denotation; 
and each of the terms M, P, S, applies to, is a name of, 

the whole (s.MJM , and each one maybe asserted of either 
of the others. Thus this construction entitles the hearer to 


the assertion S is P, which had not been actually asserted. 
The speaker who asserted first M is P, and then S is M, 

must have had before him at starting the whole fs M 

which his hearer reached as a result of putting together 
M is P, S is M. 

Thus the conditions of affirmative Mediate Inference 
with S, M, and P as terms may be formulated as follows : 
If the denotation of any two terms, M and P, is identical, 
any third Term S, which is identical in denotation with 
either of them, is also identical in denotation with the 
other. If in accordance with this pronouncement I reach 
a whole of denotation which has intensions M, P, and S, 

thus (s M P) , it is obvious that I am as much entitled to 

assert identity of denotation between S and P as between 
M and P or S and M. 

For negative Mediate Inference with Terms, S, M, P 
the following canon may be suggested : 

If of two terms S, P, one is, and one is not, identical 
in denotation with a third Term M\ then 8 and P are 
not identical in denotation. 

Hypotheticals are all either (1) Immediate Inferences 
(e.g., If M is P, P is M), or Mediate Inferences which are 

(2) fully expressed (e.g., If M is P, and S is M, then 
S is P) which like (1) may be called Self-contained; or 

(3) Mediate Inferences which are elliptical and enthy- 
mematic e.g. If M is P, S is P (v S is M), If A is B, 
E is F (v B is C, and C is D, and D is E); If S is M, 
S is P (v M is P); If S is M, S is not P (v M is not P). 

Conditionals (as distinguished by Dr Keynes and 


Mr Johnson from Hypotheticals) are merely Categoricals 
with a Subject-Term which is resolvable into Genus 
4- Differentia, expressed in Hypothetical form because the 
Predicate is limited to that part of the Class or Genus 
which is qualified or conditioned by the Differentia. 
E.g. If any triangle is equilateral, it is equiangular 
equiangularity in a triangle follows from, depends upon, 
is inseparable from, its equilaterality. A Hypothetical, 
like an Inference, must start from something given (a 
proposition or propositions). Inferences can be put in 
hypothetical form, and Hypotheticals in inferential form. 

I will examine concrete examples later, but may 
observe here that there are some propositions Categorical, 
Hypothetical, and Alternative which seem to be used as 
a rhetorical device; e.g., If Newton was not a greater 
mathematician than Kepler, the whole is not greater than 
the part, We are the victims of misunderstanding, or the 
truth is not true. These only amount to a very strong 
asseveration (1) that Newton was greater than Kepler; 
(2) that we are the victims of misunderstanding. We 
can, no doubt, and do, use prepositional forms in cases 
where instead of the/orm (S is P, etc.) resulting naturally 
from the content, the only connection of content is that 
imposed by the form (S is P, If A is B, C is D, etc.) upon 
an indifferent or even recalcitrant content : E.g., we may 

give the form S is P to an A not- A (A not- A) content, as 

in the above examples (the whole is not greater than the 
part, the truth is not true). 

In the above brief and simple statement is contained, 
I believe, the essential framework of the theory of Import 


of Categorical assertions of forms 8 is P, S is not P, of 
Hypothetical assertion, of Immediate Inference, and 
Mediate Inference. The expansion and complication 
introduced by its application to Class-Propositions will 
be considered forthwith. Here I will only remark that 
the twofold relation (affirmative and negative) of Subject 
and Predicate in assertion (which must be distinguished 
from the relation between Subject and Attribute) is quite 
disparate from the fivefold relation possible between two 
classes taken in extension, and no theory of the one can 
be made perfectly symmetrical with any theory of the 

It is, I think, to the prominence given to Class- 
Propositions and the predominant use of such Propositions 
as examples (whether symbolic or significant) in books of 
Logic, that the blurring of the clear and simple outlines 
of Assertion (affirmative and negative) is largely due. 

Such propositions as R is Q, Tully is Cicero, Courage 
is Valour, Generosity is not Justice, London is the largest 
city in the world, convert quite simply into : The largest 
city in the world is London, Justice is not Generosity, 

In dealing with ordinary class-propositions with 
quantified Subject and unquantified Predicate, the matter 
becomes more elaborate, and mistake more possible, 
because in conversion the unexpressed but implied 
Quantification (All, Some) of the old Predicate-name has 
to be expressed, since that name is now the Subject-Term; 
and on the other hand the expressed quantification (all, 
some) of the old Subject-name sinks into mere implicitness, 
that name being the new Predicate-name. Further, when 
a class-name occurs as Term without quantification, a 


different quantification is understood, when it is a Subject- 
Term, from what is understood when it is a Predicate- 
Term. E.g., in Trees are plants, Trees would be quantified 
by all. In Cedars are trees, trees would be quantified 
by some. And if we converted this last proposition, it 
would be to: Some Trees are Cedars. 

Propositions of the A, E, I, O form commonly have 
some sign of quantity attached to the Subject and not to 
the Predicate, and are said to have a quantified Subject 
and an unquantified Predicate. It has been held by 
certain reformers in Logic that all Predicates are natu 
rally quantified in thought, and ought to be explicitly 
quantified in speech. This view does not seem to be borne 
out by reflection; but careful reflection does appear to 
show that Quantification is an indispensable instrument 
of Conversion. 

The place of Quantification in Logic is very curious, 
its function being often as completely hidden from those 
whose processes of Conversion involve it, as the subterranean 
course of a train in one of the loop-tunnels of the Swiss 
Alps would be to an observer who only saw it rush into 
one opening, and emerge again in a few minutes from 
another, just above or just below. My meaning will be 
best elucidated by taking an ordinary proposition and 
tracing the changes which it undergoes in Conversion. 

Let the proposition be 

All human beings are rational (1) 

The ordinary converse of this is /[Ail] human 

Some rational creatures are human [[gome] rational] 

beings (2), \ [creatures] 


Some rational creatures are human (3). 


If I merely alter the relative position of S and P in (1) 
as it stands, and say 

Rational are all human beings, 

it is clear that Conversion in the logical sense has not 
taken place; for Rational is still the Predicate, and all 
human beings is still the Subject. The proposition has 
been merely turned round. But it may be put into the 

All human beings are rational creatures (4), 
and with this we can deal. It is not, however, any more 
than the adjectival (1), directly convertible. If altered 

Rational creatures are all human beings, 
the proposition thus obtained, besides being awkward, 
is ambiguous it is by no means clear which term is to 
be taken as Subject, and the all might even be understood 
to qualify (or quantify) Rational creatures. 

The first step towards real Conversion is taken when 
we pass from (4) to the quantificated proposition 

All human beings are some rational creatures (5). 
From this we go on to the quantificated converse 

Some rational creatures are all human beings (6) ; 
and from (6) to the unquantificated converse of (5) 

Some rational creatures are human beings (7). 
From (7) we can pass to the corresponding adjectival 

Some rational creatures are human (8). 
It is to be observed that in going from (4) to (7), we have 
not only inserted a sign of quantity before the new Subject- 
name (rational creatures) which, as the old Predicate, had 
not any to start with : we have also dropped the sign of 
quantity which the new Predicate (human beings) had 


when it was the old Subject-name. Thus, as we began 
with an unquantificated proposition, so we end with an 
unquantificated proposition. The propositions which 
logicians (on the whole) have recognised and dealt with 
are unquantificated propositions ; it is for enabling us to 
pass (by an elliptical procedure) from unquantificated to 
unquantificated propositions that the ordinary rules of 
Conversion and Reduction of Class-Propositions and 
Syllogisms are framed; it is of unquantificated propositions 
that the "nineteen valid moods" of the traditional 
Categorical Syllogism are composed. 

In converting an E proposition, we should proceed 
as follows : Let the proposition to be converted be, 
No R is Q (1). (1) = (2) Any R is not Q (by 
grammatical equivalence). Quantificating (2) 
we get, Any R is not any Q (3). (3) converts to, Any Q 
is not any R (4). By disquantificating 

(4) we reach (5), Any Q is not R. And 

(5) = No Q is R (by grammatical equi- 

My view then is that the usage of Logic and of ordi 
nary speech is on the whole to be justified, and yet that 
Quantification is possible and valid in a subordinate office, 
as a necessary transformation stage of propositions. This 
can be made clear by reference to the Import of Categorical 
Propositions. What a Categorical proposition affirms or 
denies is, identity of denotation of the S and the P in 
diversity of intension. Denotation of S and of P in an 
affirmative Categorical Proposition are the same; intension 
of the S and P being, of course, always diverse in 
propositions of the form 8 is P. And denotation is 
sufficiently indicated by the S ; identity or otherness is 


indicated by the copula (is or is not) while diversity of 
intension comes into view only when the Predicate is 
enunciated. In regard to any assertion, we want to know 
in the first place ivhat it is of which something is affirmed 
or denied; this knowledge is given with the enunciation 
of the Subject, which indicates the thing or things spoken 
of. We want, in the second place, to know what it is that 
is affirmed or denied of the thing or things indicated by 
the Subject. This information is supplied by the Pre 
dicate that is, by its signification or intension, since it 
is evident that in affirmative propositions the application 
of the Predicate is identical with, in negative propositions 
is altogether distinct from, that of the Subject. Hence it 
seems clear that in the Predicate of any proposition, it is 
intension, and not denotation, which is naturally and 
generally prominent. This is confirmed by the considera 
tion that we commonly use Adjectival Predicates, if 
appropriate Adjectival Terms are available; and that 
such terms cannot in English (though they can in many 
languages) take the sign of the plural, while the Substantive 
Terms which they qualify can, and no one doubts that the 
application of an Adjectival Term is the same as that of 
the Substantive Term which it qualifies. Now if it is the 
primary function of the S in any Categorical Proposition 
to indicate denotation, while it is the primary function 
of the P to indicate intension, it seems obvious that 
quantifying is appropriate, and may be necessary, in the 
case of S, but not in the case of P, under ordinary 
circumstances. And a further reason against admitting 
Quantification (except as a transformation stage) in most 
propositions, is deducible from the consideration that what 
propositions affirm or deny is the identity of denotation 


(in diversity of intension) of S and P; for in a quantificated 
affirmative, though indeed identity of the terms is still 
asserted (as it is bound to be), the fact that the denotation 
of both terms is made prominent tends to blur this 
especially where difference of extent of the classes referred 
to is suggested. It might indeed be maintained that 
where both terms of our propositions are taken purely in 
denotation, quantincated propositions are most appropriate, 
being the form of proposition which makes the denotation 
of both S and P most prominent. But both terms cannot 
be taken purely in denotation. If, e.g., in S is P, both 
S and P were taken in denotation only, then to say 8 is P 
would be exactly equivalent to saying S is S, for the 
denotation of P is the very same as that of S. On the 
other hand, the view here advocated of the Import of 
Categorical Propositions justifies the recognition of 
Quantification as a phase of propositions. For the 
Predicates of propositions have denotation as well as the 
Subjects, and (in affirmative propositions) a denotation 
which is identical with that of the Subjects. It is 
therefore possible, and under certain conditions allowable 
and necessary, to make this prominent by quantification. 
And the Subjects of propositions have intension ; and this 
may be allowed to come into prominence by dropping the 
sign of quantity which inevitably fixes attention rather 
upon the denotation than the intension of a term. What 
Sir Wm Hamilton hoped for from the doctrine of Quanti 
fication was, that by its help the relations of classes, 
as well as the relation of Subject and Predicate, could 
have been exactly expressed by the form of Assertion. 
But Quantification is entirely and for ever unequal to the 
accomplishment of such a task. 


The above may be further confirmed and illustrated 
by a consideration of the traditional logical treatment of 
O Propositions. Of the four Class Propositions A, E, I, 0, 
the first three have always been regarded as capable, the 
fourth as incapable, of Conversion. 

We have seen that propositions on their way to 
Conversion have to undergo the process of Quantification. 
But the reason why O (Some R is not Q) is pronounced 
inconvertible is not because there is any more difficulty 
in quantifying its Predicate than in quantificating the 
other propositions, but because, when the quantiftcated 
converse of (any Q is not some R) has been reached, the 
quantification cannot be dropped without an illegitimate 
alteration of signification. For the commonly accepted 
signification of the disquantificated converse of O (Any 
Q is not R) implies a quantification different from that 
which has been dropped the dropped quantification being 
some, the quantification understood as involved in the 
unquantificated Proposition (Any Q is not R) reached 
by dropping it, being any. And as, at the same time, 
ordinary thought and speech will not admit the explicitly 
quantificated form, it is inevitable that a Logic which 
deals with the forms of ordinary thought and speech 
should regard as inconvertible. Let us take as a con 
crete instance the Proposition, Some trees are not oaks (1). 
This becomes by quantification (2) Some trees are not 
any oaks, which converts to (3) Any oaks are not some 
trees. Dropping the quantification of (3), we get (4) 
Any oaks are not trees, and this would be understood 
to mean (5) Any oaks are not any trees (= No oaks are 
trees). (General Logic, p. 58 &c.) 


(1) All lilies (8) are beautiful (P) 
converts to : 

(2) Some beautiful things (P) are lilies (S), 
and this again converts to : 

(3) Some lilies (S) are beautiful things (P). 
Obviously the quantification some in (2) must have been 
implicit, though unexpressed, in (1); and the explicit 
quantification some in (3), must have been implicit in (2). 

It is clear that it is the quantified Subject and 
Predicate in Class-Propositions which correspond to the 
S and P in S is P. E.g., in (3) some lilies is S, and [some] 
beautiful things is P. Similarly with (1) and with (2). If 
in (1) e.g., the denotation of " beautiful " were not limited 
to the denotation of "all lilies" if, that is, All lilies (S), 
were not denotationally identical with only [some] beautiful 

things (P), \^Jj, then what the proposition asserts would 

be identity (of denotation) between lilies and all beautiful 
things an interpretation which is neither intended nor 

(1) No men (S) are angels (P) 
converts to : 

(2) No angels are men. 

Angels was implicitly quantified universally in (1), 
otherwise the explicit universal quantification of that 
term-name in (2) would not be possible. By implicitly 
quantified I mean that there is no explicit quantification 
but that explicit quantification is justified. 

In Conversion, as we have been seeing, the Subject-name 
of the converse is supplied with a sign of quantity which 
it had not at first, and the Predicate-name of the converse 
is deprived of the sign of quantity which it originally 


had. To sum up : The explanation of this change intro 
duced into Categorical Propositions, when they undergo 
conversion, is that "the natural way of thinking a 
Categorical Proposition is to emphasise the extension - 
aspect in the Subject and the intension-aspect in the 
Predicate; where an adjective of quantity is expressed, 
it is inevitable that the aspect of extension should 
have attention drawn to it. Further, the quantification 
of the new Subject-name makes clear that this name 
has had, throughout, an extensive aspect, though that 
aspect was not emphasised or explicitly brought into 
notice as long as it was a Predicate-name. The mere 
transposition of the Predicate into the place of the Subject 
could not suffice to give it extension unless it had had 
extension from the beginning, since Conversion is not a 
legitimate process if it does more than infer something 
which is true supposing the inferend is true! (Primer 
of Logic, p. 34.) 

S is P, 8 is not P, are susceptible of Obversion, and 
there is no difficulty in applying this process to Class- 
Propositions in accordance with the simple procedure 

applicable to the former. S is P, (S P) , obverts to S is not 

not-P ; S is not P, obverts to 8 is not-P. 

/All robins\ y^SX 

All robins are insect-eaters, (gomeTnsect-) (>-< ) b verts 

to No robins (S) are [any] not-insect- eaters (P) (=A11 
robins are-not not-insect-eaters). Robins being in 
cluded in the group of insect-eaters, are (by Law of 
j. 3 


Excluded Middle) excluded from all those things, whatever 
they may be, that do not eat insects : fcobinsj 

No painters are mathematicians -/P )(M\- obverts to : 

x f ^_s 

All Pcmifers (S) are not [any] mathematicians (P). 
In inference by Added Determinants of the form: If 

R is Q, then XR is Q, ^3/, it is because the relation 

of identity in extension between R and Q remains un 
affected by the intension added to the Subject, that we 
can add this intension. 

E.g. If all ices are unwholesome, then strawberry ices 
are unwholesome. 

Inference by added Determinants of the form : If 
R = Q then ZR = ZQ, as applied to quantity or number, 
depends on the principle : If equals be added to equals 
the wholes are equals. 

E.g. If 2 + 2 = 4 ........................ (1) 

then 2 + 2 + 3 = 4 + 3 .................. (2). 

Here there are in (1) two related objects or groups : 

These two are now transformed, by the addition to 
each of another object, exactly similar, into objects of 
which both denotation and intension have been modified, 
but in exactly the same way in both, so that the relation 
of equality is maintained. 

So, if 40 shillings = 2 pounds, 
then 40 shillings x 4 = 2 pounds x 4. 


Again, if 500 will buy one motor, 

then 1000 will buy two motors. 

Or, if two Northerners can tackle three Southerners, 
four Northerners could tackle six Southerners. 

I will venture at this point to quote and consider a 
passage from a little Logic book 1 which has been reprinted 
many times since it was first published in 1870, and is 
still largely used in schools and colleges, and recommended 
for examinations. The author says: "There are modes 
in which all persons do uniformly think and reason, and 
must think and reason. Thus if two things are identical 
with a third common thing they are identical with each 
other. This is a law of thought of a very simple and 
obvious character, and we may observe concerning it: 

1. That all people think in accordance with it, and 

agree that they do so as soon as they under 
stand its meaning. 

2. That they think in accordance with it whatever 

may be the subject about which they are thinking. 
Thus if the things 2 considered are 
The Metropolis, 

The most populous city in Great Britain, 
since the Metropolis is identical with London, and 
London is identical with the most populous city in 
Great Britain, it follows necessarily in all minds that 
the Metropolis is identical with the most populous city 
in Great Britain. 

Again, if we compare the three following things 2 

The most useful metal, 
The cheapest metal, 

1 Jevons Elementary Lessons in Logic. 2 Italics mine. 



and it be allowed that The most useful metal is Iron, 
and Iron is the cheapest metal/ it follows necessarily in 
all minds that the most useful metal is the cheapest. 

We here have two examples of the general truth that 
things identical with the same thing are identical with each 
other 1 , and this we may say is a general or necessary form 
of thought and reasoning. 

Compare again the following three things^ 
The earth, 
Bodies revolving in elliptic orbits." 

As far as I know I am the first person to question 
this " simple and obvious law of thought," that " if two 
things^ are identical with a third common thing 1 they are 
identical with each other." And yet it is not a law 
either of thought or of things, and it is not simple and 
obvious, on the contrary it is untrue and impossible. 
No thing can be identical with any other 
thing : London, The Metropolis, The most XtJndonX 
populous city in Great Britain, are not three /metropolis A 

r f J most I 

things, but three names of one thing. If not, V populous / 
we could not say : The Metropolis is iden- \^crty^/ 
tical with the most populous city in Great 

The explanation of this passage from Jevons, so chaotic 
when we come to examine it, is, I suppose, that like so 
many other thinkers, Jevons, in spite of all his ability 
and originality, was not clear about the different sorts 
of oneness and difference, and (as in his "great rule of 
inference " the " Substitution of Similars ") persistently 
confused together Identity of Denotation or Extension, 

1 Italics mine. Compare Hansel s interpretation of the Law of 


and Sameness of Intension, denotative one-ness, and 
qualitative one-ness. We can no more substitute 
" similars " in inference than we can " identify " one 
thing with another thing. " Interchangeability of de- 
notational identicals" would be a much better name for 
what Jevons means. 

Similarity is the category of classing, not of affirma 
tion this pencil or this stamp may be similar in the 
highest degree to that, but this is not that. On the 
other hand, this is " the man who was," but how tragically 
different. This girl is incredibly like what her grand 
mother was at 17, but I do not therefore take her for 
her grandmother, who at 17 was fair and fresh and active, 
but is now faded and infirm. I have so far learnt to 
discriminate between cases in which exact similarity is, and 
those in which it is not, evidence of individual identity. 

Similar confusion occurs in a curious form in Mill 
(Logic, I. 116, 9th ed.) when he gives as examples of 
propositions in which simple Resemblance is asserted the 
following : 

" The colour I saw yesterday was a white colour," 
" The sensation I feel is one of tightness." 

Here there seems to be confusion between assertion 
(S is P, identity in diversity) and classing (grouping of 
this instance with other instances, in virtue of resemblance 
or similarity), and complete oblivion of anything like a 
general view of import. We find a like want of clearness 
in a passage in Jevons Elementary Lessons, p. 65, when 
he says : " The proposition Gold is a yellow substance 
states stick an agreement of gold with other yellow sub 
stances that we know it to have the colour yellow," etc. 


"Mill tends to drop out of account in his treatment 
of names and propositions not only all surplusage of in 
tension beyond connotation, but also all explicit reference 
to the extension aspect. But this the application of 
names is in the very forefront of importance. For Mill, 
connotation swells and grows till it almost fills the 
picture, whether we are dealing with Terms or with 
Import of Propositions. Connotation (where there is 
Connotation) may determine application. But without 
application somehow determined, all use of names and 
terms is impossible. Mill himself seems to admit this 
when he says of Hobbes s definition of Categorical affirma 
tive Propositions ( In every proposition what is signi 
fied is the belief of the speaker that the predicate is a 
name of the same thing of which the subject is a name ), 
that it is true of all propositions and is the only account 
of import which is rigorously true of all propositions 
without exception. It is odd that Mill, while setting 
aside and belittling Hobbes s analysis, should have been 
content to furnish finally as his own contribution to the 
theory, nothing better than an analysis (and an ex 
ceedingly unsystematic one) of the imports of different 
classes of propositions. 

Hobbes, as we have seen, lays all the stress on appli 
cation of names on the denotation, not the connotation, 
aspect and this carried on into Syllogism would justify 
the fundamental importance of identity of application of 
the Middle Term (however this identity may be deter 
mined). (Cp. the requirement that the Middle Term in 
a class syllogism must be distributed. ) The same 
would hold of Immediate Inference. And it may be 
observed that Jevons doctrine of Substitution of Similars 


does really lay like emphasis on the supreme part played 
by application for the substitution referred to by the 
great rule of inference which Jevons gives, is Sub 
stitution not of Similars but of terms having identical 
application. The rule runs as follows : 

The one supreme rule of inference consists... in the 
direction to affirm of anything whatever is known of its 
like, equal or equivalent. The Substitution of Similars 
is a phrase which seems aptly to express the capacity of 
mutual replacement existing in any two objects which are 
like or equivalent [= ?] to a sufficient [= ?] degree (Prin 
ciples of Science, p. 17, 3rd edit.). 

That the substitution here referred to is in fact 
substitution of terms having identical application is 
obvious on the most cursory examination, and is ap 
parent at first sight from Jevons own examples in 
illustration, e.g. 

(a) Snowdon (1) 

Highest mountain in England or Wales (2) 

(Something) 3590 feet in height (3). 
(6) The Lord Chancellor (1) 

The Speaker of the House of Lords (2). 
(c) God s image (1) 

Man (2) 

Some reasonable creature (3). 

It hardly needs pointing out that in (a) and (c) (1), 
(2) and (3), and in (6) (1) and (2), respectively, are not 
qualitative similars, but numerical, historical, or ex- 
tensional, identicals intension is in each case different, 
but extension (and therefore application) the same. On 
the other hand, taking things that are so similar as 
to be intrinsically indistinguishable, we see at once that 


they cannot be thus substituted the one for the other. 
That house/ e.g., may be similar in the highest degree 
to another standing next it, but the one is not the other, 
and in inference could not be substituted for it. This 
copy of Giorgione s Richiesta may be an exact copy, 
yet could not be substituted for it as The highest 
mountain in England or Wales could be substituted 
for Snowdon. One of a pair of twins may be so like 
the other as to be commonly mistaken for him yet 
owing to the one having come into the world a brief 
space of time before the other, he may be the heir to 
a dukedom and inheritor of an immense fortune, while 
the other is neither the one nor the other, and to c sub 
stitute the one for the other would be inadmissible and 
even felonious" (Mind, 1908, p. 531, etc.). 

" When Jevons (Principles of Science, ch. in.) discusses 
the Import, etc., of Categorical Propositions, expressing 
them as Equations (A = B, etc.), and speaking of them as 
Identities, I find that some of his examples and some of 
his explanations are quite in accordance with my analysis. 
E.g., when he takes the Proposition, Tower Hill/is/the 
place where Raleigh was executed, and says that it ex 
presses an identity of place ; and whatever is true of the 
one spot is true of the spot otherwise defined, but in 
reality the same. But when he goes on to say that the 
same analysis can be applied to e.g., the Proposition 

(1) Colour of Pacific Ocean = Colour of Atlantic 
Ocean, finding no distinction between this and e.g. 

(2) Deal Landing-place of Caesar, except that in 
(1) we assert identity of single qualities while in (2) we 
express identity of groups of qualities, it is clear that 
there is confusion between extensional and intensional 



same-ness. The colour of the Pacific Ocean may be 
exactly similar to that of the Atlantic, but we certainly 
cannot say that the one is the other in the sense in which 
we can say that Deal is the place where Caesar landed. 
This confusion ruins Jevons whole account of inference, 
and is even betrayed by the very name-Substitution of 
Similars which he has chosen to characterise his theory " 
(Mind, 1893, pp. 450, 451). 

It would hardly be worth while to take warning 
examples from Jevons and Mill, if thinkers generally 
had outgrown this confusion between the different kinds 
of Same-ness or One-ness which has had such a devas 
tating effect upon theories of import in particular; but 
there are indications that this is not the case. 

For instance, Mrs Ladd Franklin, in discussing the 
Import of Categoricals, says : " The reason that so many 
different views are possible is a very simple one. Every 
term is a double-edged machine it effects the separating 
out of a group of objects, and it epitomises a certain 
complex of marks. From this double nature of the term, 
it follows... that a proposition which contains two terms 
must have a fourfold implication.... Whoever says, for in 
stance, that All politicians are statesmen must be prepared 
to maintain that the objects politicians are the same as 
some of the objects statesmen ; and also that the quality- 
complex politician entails the quality-complex statesman, 
and is indicative of the presence of some of the objects 
statesmen.... In other words to say that a is b is to affirm 
that both from the objects a and from the qualities a are 
inferrible both objects b and qualities 6. [But if a is b, 
objects a actually are objects b, and from qualities a, 
qualities b need not be inferrible e.g. a man may be 


a politician (a) without being a statesman (b)]. Now it 
is open to the logician to say that any one of these four 
implications is the most important or the most prominent 
implication of the proposition, but it is not open to him to 
say that less than all four of them is the complete impli 
cation 1 . Any one of the four is a sufficient groundwork 
on which to work out the entire system of reasoning" 
(Mind 1890, p. 561). 

By this I believe is meant that we may understand 
both S and P in denotation, or both in intension (con 
notation), or S in denotation and P in intension, or S in 
intension and P in denotation. But when it is said that 
to affirm a is b is to affirm that both from the objects a 
and from the qualities a, both objects b and qualities 6 are 
inferrible, I reply that such " inference " is only possible 
provided we have already understood a is b to assert 
identity of denotation of a and b (in diversity of inten 
sion). As regards the concluding assertion, I proceed 
shortly to examine " the four " and to show that not one 
of them is even possible. 

Mrs Franklin s view of the " fourfold implication of Pro 
positions in Connotation and Denotation " is approved by 
Dr Keynes who (in his Formal Logic, 3rd ed. p. 147, etc.), 
expounds the matter as follows : 

"(i) If we read the subject of a proposition in de 
notation and the predicate in connotation, we have what is 
sometimes called the predicative mode of interpreting the 
proposition. This way of regarding propositions un 
doubtedly corresponds in the great majority of cases 
with the course of ordinary thought ; that is to say, we 

1 Of course the important question is: Exactly how are "all four" 
implicated ? 


naturally contemplate the subject as a class of objects of 
which a certain attribute or complex of attributes is pre 
dicated "(P- 149). 

" (ii) Subject in denotation, predicate in denotation. 

If we read both the subject and the predicate of a 
proposition in denotation, we have a relation between two 
classes, and hence this is called the class mode of inter 
preting the proposition. It must be particularly observed 
that the relation between the subject and the predicate is 
now one of inclusion in or exclusion from, not one of pos 
session. It may at once be admitted that the class mode 
of interpreting the categorical proposition is neither the 
most ultimate, nor generally speaking that which we 
naturally or spontaneously adopt. It is, however, extremely 
convenient for manipulative purposes, and hence is the 
mode of interpretation usually selected, either explicitly 
or implicitly, by the formal logician" (p. 151). 

" (iii) Subject in connotation, predicate in connotation. 

If we read both the subject and the predicate of a 
proposition in connotation, we have what may be called 
the connotative mode of interpreting the proposition. In 
the proposition All S is P, the relation expressed between 
the attributes connoted by S and those connoted by P is 
one of concomitance the attributes which constitute the 
connotation of S are always found accompanied by those 
which constitute the connotation of P " (p. 154). 

" (iv) Subject in connotation, predicate in denotation. 

Taking the proposition All S is P, and reading the 
subject in connotation and the predicate in denotation, we 
have The attributes connoted by S are an indication of 
the presence of an individual belonging to the class P. 
This mode of interpretation is always a possible one, but 


it must be granted that only rarely does the import of a 
proposition naturally present itself to our minds in this 
form " (p. 146). 

I proceed to examine the four readings here recom 
mended to us. (See my article on Logical Judgment in 
Mind, 1893, pp. 452, etc.) 

Since, it is said, terms may have Denotation (Exten 
sion) or Connotation, or both, any Proposition of the form 
S is P may be read wholly in Denotation, or wholly in 
Connotation, or S in Denotation and P in Connotation, or 
S in Connotation and P in Denotation ; thus giving four 
possibilities. If there can be four valid formal theories of 
Assertion, since each differs considerably from the others 
it ought no doubt to be possible, as Mrs Ladd Franklin 
affirms, to have four systems of Logic corresponding to 
those four theories respectively. It would indeed be 
interesting to have even the most meagre outline of even 
one of these four possible theories. But leaving this point, 
let us look at the alternative readings of S is P pro 
positions which are here formulated. That these four 
alternatives are possible, or indeed that any of them 
is so, I most emphatically dispute. If the assertion 
expressed by S is P is to be read wholly in Extension, 
(1) then since the Application of S is (by the force 
of the copula) identical with the Application of P, 
if we ignore the element of Connotation or Intension 
(in which alone there is difference) we must express 
the assertion as S is S. S is not P is clearly not capable 
of being even supposably read in Extension only, since 
diversity of Signification in Subject and Predicate is 
rendered indispensable by the negative copula. 

(2) If S is P is to be read in Connotation (or Inten- 


sion, or Comprehension) only, again the affirmative S is P 
must melt (cp. Lotze) into 8 is 8; for how can any 
connotation be any other connotation ? If it is said that 
S is P expresses a combination of the connotations of S 
and P, it seems sufficient to point out that the only way 
in which connotations can be combined is by co-existing 
in one extension. 

Again, if in S is P (3) S is taken in Denotation only, 
and P in Connotation only, or (4) S in Connotation and 
P in Denotation, what is the force of is ? Between what 
is Identity supposed to be asserted ? We can no more 
say that Denotation is Intension than we can say that 
This kitten is Animality. Is it not plain that, for is to 
have any assertive force, there must be denotational 
Identity between S and P (in S is P), and that for any 
significance to attach to the assertion, there must be a 
diversity of Connotation or Intension ? 

In order that e.g., All R (= S) is [some] Q (= P) may 
be interpreted (" in Extension ") to mean : 

Class R/is /included in class Q ; /ciass>\ 

/ [some] in- \ 

or (" in Connotation ") to mean : I cluded in / 

\Class Q/ 

Attributes R I are I accompanied by Attributes Q; 

not only must both aspects have been taken account of in 
both Subject and Predicate of the original Proposition but 
the interpreting Propositions are unintelligible without a 
similar analysis having been applied to them as they 
stand, and the interpretations into " in Extension " and 
"in Connotation" are seen to be entirely founded not on the 
form of the propositions, but on the intensions of the sub 
ject and predicate. Granted that All R is [some] Q (1), 
may mean Class R is included in Class Q (2); this 


(2) again has got to be analysed as denotational identity 
in intensional diversity. Unless so understood, is is not 
admissible, for in intension Class R and included in Class 
Q are diverse, we could not say that in intension the one 
is the other. So it is the denotation of Class R that is 
identical with the denotation of the intensionally diverse 
predicate, [some] included in Class Q. 

The Identity-in-Diversity analysis starts simply and 

solely from what is asserted, the whole 

And the analysis (identity-in-diversity) being abso 
lutely general and highly abstract, a mere skeleton 
analysis, admits of further determination of various kinds ; 
the only proviso is, that these further determinations start 
from and presuppose the skeleton analysis. 

E.g. in : All Isosceles Triangles have the angles at the 

base equal 
(= All Isosceles Triangles are having the angles 

at the base equal), 

the identity-in-diversity analysis offers no obstacle to the 
view that the intension of the Predicate is inseparable 
from that of the Subject. It allows indeed of this being 
recognised in the fullest way. In fact the inseparability 
of the intension of P from that of S quite inevitably 
involves identity of denotation (in diversity of inten 

Mr Russell in Mind, 1905, proposes to substitute for 
Frege s analysis of Categoricals (of which in 1903 he 
approved) a very complicated statement e.g. instead of 
understanding "The father of Charles II was executed" to 
express identity of denotation (Bedeutung) in diversity of 
intension (Sinn), he would interpret it as follows : " It is 


not always false of x that x begat Charles II, and that 
x was executed, and that If y begat Charles II, y is 
identical with x is always true of y" (p. 482). 

As regards this I would point out that in my view 
(1) the speaker who asserts that The Father of Charles II 
was executed starts from the subject-matter of assertion, 

/Father of\ 

the complex whole : Charles n 1 and in order to deal in 

V Executed / 

any way with this, it has to be first of all analysed on the 
identity in diversity plan, so that " Father of Charles II " 
and " executed " are referred to the denotation of the 
subject as its intension. (2) The statement : " It is not 
always false of x, etc." involves several repetitions of 
identity in diversity : 

(a) It is not always false of x t (b) that x begat Charles 
II and that (c) x was executed, and that (d) if y begat 
Charles II, (e) y is identical -with x t (f) is always true 
of y. 

Unless these clauses are to be understood as identities- 
in-diversity, what can be made of them, what is the 
connexion between their elements ? I understand that 
Mr Russell s object in giving up Frege s view and putting 
forward this complicated substitute, is to eliminate " de 
noting phrases" and so get rid of inconvenient implications 
of " existence." But he does not seem to have escaped 
identity of denotation in diversity of intension (in the 
ordinary sense of denotation and intension) and I cannot 
avoid the conviction that any form of proposition is in 
capable of determining questions of " existence," just as 
much as forms of proposition (8 is P, etc.) are incapable 


of completely and determinately expressing relations of 

In: All Cavicornia are Ruminants, 

All Antelopes are Cavicornia, 

All Antelopes are Ruminants, 

the relation of Terms may be diagrammatically represented 

The true Middle Term is the some Cavicornia of the 
Minor Premiss ; for it is only that part of the denotation 
of Cavicornia which is common to both Antelopes and 
Ruminants, that is the bond of connexion between them. 
The Ruminants that Antelopes are, are the Ruminants 
whose denotation is identical with that of those Cavi 
cornia that are identical with Antelopes. Of those 
Ruminants whose denotation does not coincide with that 
of any Cavicornia, and of those Cavicornia whose denotation 
does not coincide with that of any Antelopes, it must be 
said that they are not Antelopes, and that Antelopes are 
not they. It is the indefiniteness of the some by which 
Ruminants in the Major Premiss, and Cavicornia in the 
Minor Premiss are implicitly quantified, that makes it 
necessary to sweep in the whole extension of Cavicornia, 
so as to make sure that those Cavicornia with which (as 
being Antelopes) we are concerned, are Ruminants. 

Sameness of Denotation (identity) of Middle Term in 
Mediate Inference is that which connexion between Major 
and Minor Terms must depend upon, for it cannot depend 
on sameness of intension or exact similarity (cp. Jevons 
"Substitution of Similars")] intensional sameness, the 


closest similarity, would not justify substitution if it 
would, there would be no reason why the Middle Term in 
a Syllogism should be distributed the intension M would 
be all that could be required as a link, 
and (as in all S is M, all P is M), S 
might be identified with one part of the 
Class M, and P with another part, and as a 
result S identified with P, which is absurd, 

No diamonds are red 
This stone is red. ( D 

/. This stone is not a diamond. 
No diamonds = P 
[any] and [some] red = M. 
This stone = S 

Some red is, denotationally, part of any red. 

All sapphires are blue 
This stone is not blue. 
. . This stone is not a sapphire. 
A II sapphires = P 
[any] and [some] blue = M. 
This stone = S 

Some blue is identical with part of any blue. 
It is some red and some blue which are the true middle 
terms. Compare the account of the "Antelope" syllogism 

It is because Class relations as expressed in the A, I, O 
forms are indeterminate, that in Mediate Inference we 
cannot make the Terms correspond exactly with the clear 
and perfectly definite forms of the 8 is P, S is not P type 
when we are dealing with unquantified class-syllogisms. 
J. 4 



London : 
Capital of 

This may be done however in the case of what has been 
called Traduction, where all the subjects are singular and 
have identical denotation, e.g. 

London is the largest city in the 


London is the capital of England, 
The capital of England is the 

largest city in the world. 
It is done exactly in every Mediate Inference (Traduc- 
tional or other) in which the denotations of all the Terms 
are determinate, e.g. 

The Syndics and Night Watch are 
two of Rembrandt s masterpieces; 
The Syndics and Night Watch are 
two of the pictures in the New 
Museum at Amsterdam; 
Two of the pictures in the New 
Museum are two of Rembrandt s 

It seems unnecessary here to consider the differences 

of Syllogistic Figure, and of Mood in as far as variation 

of Mood in Class- Propositions goes beyond the three cases 

possible when we use the S is P, S is not P forms only 

i.e. (1) M is P 


(M P S] 

ight Watch 

Two of R. s 

Two of the 

pictures in 

the New 




P is not M 
S is not P 

S is not M 
S is not P 



Such differences of Mood and Figure may result from 
the indeterminateness of A, I, O, and further variations 
of determination due to the fact that the some of ex 
plicit quantification is itself indeterminate. If we allow 
conversion of Class-Propositions to be possible, we must 
admit that in every case the Terms are either explicitly 
or implicitly quantified; owing to the conventions of 
customary speech, the quantification is generally im 
plicit ; when made explicit it is mostly indeterminate ; 
but its possibility is incontestable proof of the denotation 
of Predicates. If in affirmative Categoricals it were 
possible (which it is not) to simply add the intension of 
the Predicate to the denotation of the Subject, we should 
avoid all difficulties due to the implicit some and the in 
determinateness of denotation of Predicate ; but then the 
Propositions would be incapable of Conversion. 

An examination of concrete Hypothetical, Conditional, 
and Disjunctive (Alternative) Propositions shows that 
here too the analysis of Categorical Affirmation as identity 
of denotation in diversity of intension is 
applicable. Take the following Conditionals : 
If any child is spoilt, he is troublesome, 
asserts the identity of denotation of spoilt 
child with troublesome child. 

If any rose is blue, it is a curiosity, asserts denota- 
tional identity of Blue Rose with a Curiosity. 

Take the following Hypotheticals, of 
which (1) is Self-contained, i.e. the conse 
quent is a necessary consequence of the 
antecedent taken alone : 

(1) If all men are fallible and the Archbishop is a 
man, the Archbishop is fallible. 




What is asserted is, that granting that the denotation 
of man is part of the denotation of 
fallible, and that the denotation of the 
Archbishop is part of the denotation of 
man, then it follows that the denotation 
of Archbishop is part of the denotation 
of fallible. 

(2) If Charles I had not deserted Strafford, he would 
be deserving of sympathy. 

This asserts that supposing denotation of Charles I 
to be identical with denotation of one 
who did not desert Straff ord, then (be 
cause not to have deserted Stafford 
would have been to deserve sympathy) the 
denotation of Charles I would have been 
the denotation of one deserving of sym 
pathy. In this example it is not from the expressed 
antecedent alone that the consequence follows, but from 
that antecedent taken in conjunction with another (un 
expressed) proposition. 

(3) If the building goes on, he will not recover. 

This may be expanded into : 

If the work goes on, great noise will be made ; 

If great noise is made, he will be disturbed by it ; 

If he is disturbed, he will not sleep ; 

If he does not sleep, he will die. 

The conclusion he will die results from a series of 
suppositions in which building going on (1) is identified 
(denotationally) with making noise (2), making noise with 
disturbing him (3), disturbing him with preventing his 
sleeping (4), preventing his sleeping with preventing his 
recovery (5). What holds the argument together deno- 



tationally is just as much of the denotations of (2), (3), 
(4) and (5) as are identical with 
the denotation of (1). (This is not 
affected by the circumstance that 
here denotational identities follow 
from intensional connexions). 

The efficacy of the identity- 
in-diversity analysis is I think 
nowhere more strikingly seen than 
in its application to Hypothetical, 
especially Hypotheticals of the elliptical and often com 
plicated sort which we so commonly employ, and of which 
the illustrations (2) and (3) examined above are instances. 
I will here take as one more illustration, Lewis 
Carroll s " Logical Paradox," the discussion of which has 
at intervals amused the readers of Mind since 1894 1 . 
The case presented by Lewis Carroll is, that in a certain 
barber s shop there are three attendants, Allen, Brown 
and Carr, and at no time are they all out together, 
i.e. Allen or Brown or Carr is always in (1). According 
to this we may have A, B and C all in, only A and B in, 
only A and C in, only B and C in (a), only A in, only B 
in (b), or only C in ; and 

(1) all times are times at 
which one man is in. But 

(2) if Allen is out Brown 
is out (because Allen 
has been ill and cannot 
go out without Brown). 

So (a) B and C are in and Allen is out, and 

(b) B is in and Allen and Carr are out, are in- 
1 See Mind for 1894, 1895, 1905. 

when A is 


Times when 
A and B are 

Times when 
A and C are 


Times when 

A and B and 

C are out 

54 A NEW 

admissible cases (a) is barred by (2) because A is out 

implies B is out (=A11 times 

that Allen is out are times 

that Brown is out). So Brown 

and Carr cannot be in when ( ^^wh^ ) ( > ( c ) 

Allen is out. And Allen is 

out = Allen and Brown are 

both out (by (2)). 

So in (b) Carr and Allen are out = Carr and Allen 
and Brown are out (c), and by (1) All times are times 
when A or B or C is in. So (b) as it stands is barred by 
(2), and as amended to (c), is barred by (1). 

The interest of this analysis of Lewis Carroll s instance 
is that the whole case is subject to two conditions : 

(1) That A or B or C must always be in; 

(2) That A cannot be out without B ; and these may 
conflict, and it is not easy at first sight to see exactly 
how to combine the fulfilment of both conditions, and 
exactly what denotational identities are justified by the 
combination. As in all elliptical Hypothetical, when the 
argument is expanded to a full statement the whole con 
ditions need to be explicitly taken account of; and as has 
been indicated, the whole argument in any case may be 
completely set out in a series of propositions asserting 
identity-in-diversity 1 . 

In any concrete case in which it is possible to assert that : 
If A is true, the truth of C follows, 
If A is true, the truth of C does not follow, 
it will be found on examination that either each Hypo 
thetical is elliptical or A is itself contradictory. 

1 An interesting solution of Lewis Carroll s Paradox " is offered by 
Mr Bertrand Eussell in Mind for 1905 (pp. 400, 401). He says that he 


E.g. in : 

If this is Inference, the conclusion is contained in 

the premisses ; 
If this is Inference, the conclusion goes beyond the 

it is clear that the Hypotheticals are elliptical. 

Disjunctive (or Alternative) Propositions are equi 
valent to Hypotheticals or Conditionals, and may be 
analysed in the same way. 

E.g. Any topaz is pink or yellow 

= Any topaz is pink or (if not pink) is yellow 
= If any topaz is not pink, it is yellow. 
They must come some other time than Saturday after 
noon or I cannot receive them, may mean : 

Saturday afternoon is a time when I shall be away 
from home ; 

considers the paradox to be " a good illustration of the principle that a 
false proposition implies every proposition. Putting p for Carr is 
out, q for Allen is out, and r for Brown is out, Lewis Carroll s 
two Hypotheticals are : 

(1) q implies r. 

(2) p implies that q implies not-r. 

Lewis Carroll supposes that q implies r and q implies not-r 
are inconsistent, and hence infers that p must be false. But as a matter 
of fact q implies r and q implies not-r must both be true if q is 
false, and are by no means inconsistent. Thus the only inference from 
Lewis Carroll s premisses (1) and (2) is that if p is true, q is false, 
i.e. that if Carr is out, Allen is in. This is the complete solution of the 

But (i) if q implies r and q implies not-r are not inconsistent, 
how do we know (on the above reasoning) that q is false ? (ii) We seem 
to admit here both that the truth of q implies r (1), and also that the 
falsity of q implies r. (iii) In (1) q implies r unconditionally, in (2) the 
implication is conditional on the truth of p. 


A time when I shall be away from home is a time 
when I cannot receive visitors; 

. . If they come on Saturday after 
noon they come at a time 
when I cannot receive visitors. 

In regard to the interpretation of 
Alternative Propositions, the one ques 
tion in dispute regarding the alter 
nants of the proposition is : Are they exclusive or un- 
exclusive ? Though there has been great division of 
opinion among logicians on this point, and though there 
are Alternative Propositions such as : " He came in either 
second or third," " We start either Wednesday or Thurs 
day/ in which it is quite clear that while we cannot deny 
both alternatives, neither can we assert both, yet there 
can be no doubt that in such cases the exclusiveness of 
the alternatives is due not to the form of proposition, 
but to the nature of the cases in question. " It thus 
seems that the only account which we can give of the 
general or formal import of Alternatives that is to say 
of the import which is common to every one of them 
is that if we deny one alternative, we must affirm the 
other. It should be observed that although terms used 
as alternatives are not necessarily exclusive in extension 
or denotation, they are exclusive in intension, in as far as 
they are not tautologous (in which case the alternation 
seems to vanish). Thus, in All his parishioners are 
criminals or paupers, the alternatives, though not de- 
notationally exclusive since the same parishioner may be 
both criminal and pauper are necessarily exclusive in- 
tensionally, since we cannot say that they are synonymous." 
(Primer of Logic, p. 26.) 


As regards the doctrine of Opposition, it is of par 
ticular interest from my point of view, because on the 
identity-in-diversity analysis it presents in a clear com 
pact form the equation of the fivefold class-relation to 
the traditional fourfold schedule of class-propositions. 
Any two classes indicated by intension or by symbols 
may have one of five extensional relations to each other. 
Let us take R and Q to symbolise two classes. The 
scheme may then be set out as follows : 


(1) (2) (3) (4) (5) 

A. All R is Q = (l)or (2). 
E. No R is Q = (5). 
I. Some R is Q : (1) or (2) or (3) or (4). 
O. Some R is not Q = (3) or (4) or (5). 
Of these propositions A and E may both be false, but 
they cannot both be true ; I and O may both be true, 
but they cannot both be false ; of A and 0, and of E 
and I, one is true and the other false. If A is true, I is 
true, if I is false, A is false ; if E is true, is true ; if 
is false, E is false. Reference to the diagrams makes 
the whole scheme at once self-evident, and the diagrams 
exhibit the identity or non-identity of Subject and Pre 
dicate in every case. To take concrete examples of Contra 
dictories (A and 0, E and I), we may say: 

Either all beliefs are true ((1) or (2)), or some are 

not true ((3) or (4) or (5)). 
Either no men are perfectly happy ((5)), or some are 

so ((1) or (2) or (3) or (4)). 
Conditionals come under the same rule as Cate- 



goricals in respect of Opposition, and any proposition of 

the form : If A , then C, contraposits to // not-C, then not- A, 

and may be contradicted by : the 

truth of C does not follow from 

the truth of A i.e. though A is 

true C is doubtful (which exactly 

expresses the relation between I 

and A, O and E). To take a 

concrete case : 

(1) If money go before, 

all ways lie open, 
may be contradicted 

(2) Though money go be 

fore, it does not follow that all ways lie 
open. (Primer of Logic, p. 37.) 

The Identity of Denotation in Diversity of Intension 
analysis applies whether we are considering what is 
asserted ; or the assertor the speaker or teacher who 

starts from the whole (si>) ; or the hearer or learner, who 

receives the assertion piecemeal, and finishes with the 

The two different attitudes afford some explanation of 
different theories of import of propositions, etc. It is 
plain, e.g., that the account of judging according to which 
it consists in putting two ideas together, and the Canons 
of Syllogism : (1) Two terms agreeing with one and the 
same third term agree with each other, (2) two terms 
of which one agrees and the other does not agree with 
one and the same third term, do not agree with each 
other: are adapted to the point of view of hearer or 


learner ; while the view of Brentano and Hillebrand, that 
in any judgment S is P, one object fay is present to 

the mind, is evidently appropriate to the point of view of 
teacher or speaker the hearer has to build up the whole 
he reaches it in the end, he does not start from it. 

The speaker who has before him a whole composed of 
parts of denotation e.g. a division or classification of 
Triangle, thus : 


Equilateral Isosceles Seal 


or a clockmaker with a clock, or a schoolboy with a knife, 
or an astronomer contemplating the planetary system, or 
a General in a campaign with a plan of operations sketched 
out in his mind all these can communicate to others 
piecemeal as much as they wish of that which is cognised 
by them, by means of propositions of the forms 8 is P, 
S is not P. No doubt if they can set before their audience 
the actual table of classification, the actual piece of 
mechanism, the actual knife, a working model of a 
planetary system and so on, the exposition is immensely 
helped, or may even be rendered unnecessary. Of course 
such helps are used in teaching wherever possible 
blackboard-drawings, models, lantern slides, etc. 

The difficulties of (1) impersonal and (2) elliptical 
propositions, such as (1) It rains, (2) Fire! Wolf! are 
very much mitigated if it is remembered that in every 
case the speaker must start, not from the words of his 
pronouncement, but from the matter of fact, not from the 
expressed assertion, but from what is asserted. The corre- 


spondence of the verbal assertion to that which it asserts 
is often regarded as artificial, and the verbal expression is 
called a " verbal device," or by some name that has an 
equally opprobrious implication. Of course it cannot be 
denied that if I have before me a red rose and assert : 
This rose is red, my spoken assertion consists entirely of 
words, and in particular contains a copula to which the 
red rose seems to present nothing even remotely corre 
spondent. Accordingly some logicians wish to reject the 
copula, and some think it a verbal device in the very 
worst sense, a useless, embarrassing and unjustified re 
dundancy of expression. From my point of view, how 
ever, all this is mistaken. If what is asserted in any 
S is P is identity of denotation in diversity of intension, 
then in asserting it we want not only the diverse terms 
with their intensions and denotations, but also something 
which indicates and conveys to the hearer the identity 
of denotation between the terms, and this function the 
copula is admirably fitted to perform. And the negative 
copula is just as well suited to its particular task. In 
fact the copula seems to me a very economical and 
effective means of carrying out a delicate and indis 
pensable part of the whole function of communicating by 
means of speech. It is one of the many instances in 
which men " have builded better than they knew." 

In such propositions as : The round-square is non 
existent, we cannot dispense with a one-ness of denotation 
(extension) in the subject, because, without this, round 
and square would have simply their intensional diversity 
there would be no even hypothetical joining together of 
round and square, no problem, no difficulty, no reason to 
assert "non-existence," to raise any question. Since in 


space, as known to us, roundness cannot be square, and 
squareness cannot be round, the denotation to which the 
two qualifications are assigned can " exist " only in the 
universe (or region) of hypothesis or supposition. This 
hypothetical combination is denied a place in the "uni 
verse " of actual space. 

Where intensions, attributes, are (1) incompatible, or 
(2) inseparable, then the attempt (1) to combine them in 
one subject, one denotation, as round-square, or (2) to 
separate them, as equiangularity from equilaterality in a 
triangle, is an attempt which can never be realised. We 
may "suppose" the conjunction (or separation), we can 
assert it, and trace its consequences, but that is all, as I 
might suppose that I could fly like an eagle, swim like 
a fish, and be stronger than an elephant, and deduce 
various things that I could do on these suppositions. 

In using impossible combinations as Subjects (or 
Predicates) of Propositions, or a Subject which has a 
Predicate which cannot co-inhere with it in one denota 
tion, we are perhaps sometimes simply extending forms 
and processes of language, appropriate in some cases, 
to cases to which they are not primarily and directly 

Suppose I say: 

No roses are blue rBJ KB j , 
this may be expressed also as : 

There are no blue roses, or Blue roses = 0, or 
Blue roses are non-existent. 

All these seem admissible ways of expressing the matter 
of fact indicated by the diagram MM Mn . 


Apply this to the round-square case : 

(1) No squares are round Cs j M-M . 

There are no round-squares. 
Round-squares = 0. 
Round-squares are non-existent. 

Even on this view, however, we have to postulate the 
conjunction of round and square in a suppositional de 

In speaking of Squares and Rounds in (1), we are 
naturally understood to be referring to the region or 
universe of space as known to as, by Rounds and Squares 
we mean plane figures of a definite familiar shape. 

But when we say Round-squares do not exist we 
assign only our Predicate to that same 
extended universe, and the Subject 
which is round and square belongs to 
a region of the merest, and we may even say wildest, 
hypothesis a region entirely separate from the region in 
which squares that are merely square, and rounds that 
are simply round, have their " existence." 
The round-squares are declared to be non 
existent, they are identified (in denotation) 
with something that is non-existent. 

But that non-existence does not signify complete and 
unmitigated non-existence, but only the absence of spatial 
existence in talking about round-squares we are talking 
about something, although it is an incoherent and un- 
realisable something. 

Whatever is thought of as having denotation, is thereby 
thought of as having " being," existence of some sort, of 
^uhat sort has to be fixed by intensional determination. 


All the wheels that go to Croyland are shod with 
silver, was a picturesque way of saying that 
no ordinary work-a-day wheels did ever go to 
Croyland. The wheels that went there were 
shod with silver, that is to say, they belonged 
to the same region as silver-shod wheels, viz., 
the region of imagination shod with india-rubber would 
probably have been an even more far-fetched idea, at the 
time when the saying was framed but fens have been 
drained and roads constructed, no doubt the rubber tyres 
of motors have found their way to the ruins of the ancient 
abbey, and it might some day occur to a cranky millionaire 
going in that direction to have the tyres of his wheels of 
silver metal now so much less precious than formerly, 
if only to illustrate the legend. 

" Existence " of some sort we must attribute to every 
thing of which we speak. But no particular kind of 
existence can be implied by forms (such as S is P) which 
propositions that deal both with the " real " world and 
with mere fancy or fiction, have in common. The kind of 
existence anything has is shown by the predicates we 
can give it. Any proposition S is P that I assert, is an 
entity, has some sort of existence. But the important 
question is, What sort ? Is it true, for instance ? Well, this 
must be tested by criteria. I cannot doubt (1) what is 
self-evident, as that a whole is greater than its part ; 
or (2) what is to me matter of direct experience, as 
that that flash of lightning was followed by a clap of 
thunder ; or (3) what is logically deduced from that which 
is accepted as true, e.g. if twenty shillings are equal to 
1, forty shillings are equal to 2 ; or (4) what is in 
harmony with all which I accept, as that parallel lines do 


not enclose a space ; or (5) that which is implied in what 
is accepted as true e.g. that propositions of the form 
S is P (8 is not P) (by the help of which alone the 
Laws of Contradiction and Excluded Middle can be 
asserted, and in which, in fact, most of our assertions, 
whether self-evident or disputable, must be affirmed or 
denied, supported or called in question) that proposi 
tions of these forms are possible and valid. 

As already insisted on, what S is P asserts, is that the 
denotation of S, whatever it is, is the denotation of P. 
If we start with an S which has not any denotation to 
begin with, nothing can ever bestow that which is lacking. 
But of what sort the denotation of S is, is settled by its 
intension, and the intension of its predicate, and by context, 
as in the old-fashioned school-room game in which one 
person thinks of a thing, and another person has to try 
and find out what it is by asking questions, to which the 
answer must be Yes or No. The thing questioned about 
is thought of by the questioner as being something, as 
having some existence, but of what sort it is, in what 
region it is, is revealed to him only when he knows what 
predicates, what intension, can be assigned to it. As to 
"Real" Existence, it is subject to as much ambiguity as 
Identity is, and the ambiguity in this case is far more 
difficult to clear up. How are we to define or describe 
Reality ? What about the future, what about the past ? 
The roses that have faded and fallen this year, and those 
that will blossom next year? What about ideas of the 
non-existent, which become operative in the world of 
Time and Matter? 

What are we to say of the ideal of an architect, painter, 
poet, novelist, reformer, which guides the action of the man, 


and leads to physically embodied results which may be 
widely influential ? Or even of the delusions of a madman, 
which are intensely " real " to him and may lead him to 
realise the most disastrous actions in the everyday world 
of space and time ? I remember reading a tragic story of 
the Captain of a ship who on a voyage went out of his 
mind. He fell under the delusion that various members 
of the crew were conspiring to mutiny, and with marvellous 
caution and cunning, induced first one and then another 
of the officers and men to share his suspicions of some 
of their number, and help him to secure them. He 
succeeded so well that most of the crew (it was not a large 
one) were overpowered one by one, and bound and made 
helpless. Then, having laid all his plans with superhuman 
ingenuity, with the strength and fury of a madman and 
armed with weapons which he had secreted, he fell upon 
the unfortunate victims, and the ship arrived in port 
with the Captain a raging lunatic and most of the crew 
murdered. What view are we to take of "reality" in 
such a case ? 

Or again of the perverted judgment of a dipsomaniac, 
or of such a mother as the one in The Green Graves of 
Balgowrie, which leads to cruel ill-treatment of the children 
of the person so afflicted ; or the " fixed idea " of an old- 
fashioned miser who leaves his unfortunate sons and 
daughters half-starved and uneducated, to save a lawyer s 
fee, draws his own will, with the result that it does not 
carry out his intentions, and himself dies of starvation. 

The finding of the North Pole by Dr Cook, and the 

near approach to the South Pole by Lieutenant Shackleton 

were, some months ago, on the same level of " reality " as 

far as the general public knew, and neither achievement 

J. 5 


could have been even discussed or questioned, unless it 
had been provisionally credited with "denotation," 
"existence," or "reality," in the region at least of 
supposition. We identify the denotation of P with the 
denotation of S just the same whether we merely suppose 
$ is P, or question it, or affirm it, or consciously suspend 
our judgment. 

The reproach of unreality is, it would seem, only 
pertinent when one kind of reality is mistakenly iden 
tified with another kind. 

I hope that I have in the foregoing pages made good 
my undertaking, and shown that the substitution for the 
old Law of Identity, A is A, of the principle that Every 
Subject of Predication is an Identity (of Denotation) in 
Diversity (of Intension), does provide the explicit recog 
nition and justification of S is P, S is not P propositions 
which Logic has hitherto needed but not had, and does 
furnish Formal Logic with a real and obvious basis, and 
an adequate constructive principle. 

My scheme, I hold, elucidates (among other things) 
the relations of Denotation (Extension) and Intension ; 
the general Import of Categorical Propositions and their 
relation to Conditionals, Hypotheticals and Alternatives ; 
Immediate and Mediate Inference ; the relations to each 
other, and to logical science, of the three Laws of Thought ; 
the meaning and place of Quantification; the general 
relation between Relative and Non-relative Propositions ; 
the fundamental difference between the relation of Subject 
and Predicate in Assertion and other relations which have 
been sometimes confounded with it e.g. the relations of 


(1) Subject (Substance) and Attribute, and (2) relations 
of Classes; the difference between extensional one-ness, 
and qualitative one-ness. 

On my principles, as I think, the whole of Formal 
Logic becomes a systematised and harmonious whole, with 
a sound basis, an obvious and all-pervading principle, and 
a simple and coherent structure. 




It remains to say a word about Fallacies Fallacies 
may be brought into a simple connexion with the Identity 
in Diversity analysis of Categoricals by the consideration 
that all fallacy consists in either identifying what is 
distinct or distinguishing what is identical, so that we 
get a primary division of Fallacies into (a) those of mis 
taken distinction, which are Fallacies of Tautology, and 
(b) those of mistaken identification, which are Fallacies 
in which there is failure of continuity, and may be called 
Fallacies of Discontinuity. The classification possible on 
these lines is summed up in the following Table. (See 
Primer of Logic.} 




l : a 

1 t 




M C 


<H 3 


JS c rOS 
J 8^2 

g S " S, c 

15 I ^1 

eg g 

HH >H g 


If d 


fil Ml 

1" Jils j 



<1 9 

* be 






Proposition] _ 

A ,. > I use in the same sense, as denoting the 
Assertion } 

statements of Categorical, Inferential (Hypothetical), or 
Alternative form, e.g. S is P ; If A is B, C is D ; C is D 
or A is not B. In as far as the words in an assertion can 
be considered apart from what is asserted I would use the 
name verbal expression or, if more convenient, sentence. 

By ivhat is asserted (or the assertion) I mean that 
matter of fact, or belief, or state of the case, which the 
assertion or proposition sets forth in words. The speaker 
apprehends, or is conscious of, something which he conveys 
by means of words to his audience ; that something is what 
is asserted, e.g. he sees the door open and conveys the 
fact in the assertion: "The door is open"; or he is aware 
of feeling very chilly, and conveys the fact by asserting 
the proposition : " I am chilled to the bone," or he believes 
that "Twice two is four" and asserts it. The assertion 
or proposition is of course a "verbal device" (though 
a necessary and indeed an indispensable one) and in 
particular the copula is sometimes accused of being a 
device in a specially bad sense and of having nothing 
corresponding to it in what is asserted. Such objections 
seem beside the mark no one attempts to deny the 
difference between what is asserted and the assertion of 
it the point is : Does the assertion made by the speaker 
convey to the hearer a knowledge of what the speaker 
asserts ? If so, it answers its purpose fully and perfectly. 
The copula in particular seems to me one of the most 
admirable of human devices briefly and simply and 


modestly helping to fulfil the function of conveying to 
every hearer the information, the matter of fact, the 
somewhat asserted, which any speaker desires to com 

The distinction which I draw between the attitude 
of speaker or teacher on the one hand, and hearer or 
learner or seeker on the other, which is of great interest 
and importance in Logic, seems to be specially enlightening 

Denotation (Extension) of a term means the sphere 
of its application the things of which the term is the 
name, the things to which the term applies. 

Intension of a term means the properties of the things 
to which the term applies it " may be used to indicate 
in the most general way the implicational aspect of name " 

Connotation is that part of the Intension of a Term 
which is set out in the Definition, and on account of 
which the name is applicable. 

Sameness (1) one-ness of denotation, identity; 
(2) similarity, resemblance, likeness, qualitative one-ness 
(Same-ness might conveniently be restricted to qualitative 

Identity denotational one-ness, existential or exten- 
sional unity antithetic to Distinctness, Otherness. 
Compare " mistaken identity." 

One-ness = antithetic to Difference = 

(1) denotational one-ness, identity ; 

(2) qualitative one-ness, same-ness. 

Unity = (1) Identity, (2) Same-ness, (3) any system, 
or whole made up of parts, and (4) the relation between 
such parts. 


Difference = (1) Distinctness or Otherness, such as the 
difference between this new shilling and that new shilling 
of the same minting ; (2) Diversity e.g. such a difference 
as there is between justice and generosity, humanity and 
mortality, or between an egg and the robin into which 
it hatches; (3) (Differentia) the characteristics by which 
any sub-Class (or species) is distinguished (differenced) 
from the rest of its wider containing class (or Genus). 

Diversity see Difference. 

Distinctness see Difference. 

Otherness see Difference. (Compare: give me another, 
give me a different one.) 

Similarity. There is similarity between two things 
when they resemble each other produce impressions 
which we call like ; and there is similarity between the 
different phases of one thing in so far as it remains 
unaltered. Similarity may be slight and partial, or so 
great as to amount to what has been called indistinguishable 
resemblance (= qualitative one-ness). Similarity (Resem 
blance) is antithetic to Diversity. 

The phrase exact similarity as equivalent to qualitative 
one-ness is sometimes objected to on the ground that, 
e.g., squareness or snow-whiteness or mortality have 
extensional as well as intensional one-ness that the 
mortality of Socrates is identical with the mortality of 
Newton or even that extensional and intensional one-ness 
in such cases coalesce or are indistinguishable This 
appears to involve a monadistic existence of qualitative 
or conceptual entities. I think that granting such entities 
if we could assert of any one of them, S, that it is P, we 
could not do this without postulating or implying that 
it is an identity-in-diversity. For S is given as intension- 


ally diverse from P that is, intensionally S is not P. 
If, therefore, S is P, the one-ness indicated must be 
something different from intensional one-ness. It must 
be a one-ness of being to which the diverse intensions 
are referred that is an extensional one-ness. Under 
no other condition can one-ness of intensional diversity 
be asserted. Even in the case of, e.g., obtuse-angled 
triangularity, we cannot say that Triangularity is Obtuse- 
angled-ness, but only that a triangle may be obtuse- 

The relation of Identity-in-Diversity of Subject and 
Predicate in Predication (1) must be distinguished from 

(2) relation of Subject and Attribute (the subject 
of Predication may be the attribute of a Subject 
(Substance) e.g. Triangularity is a property of plane 

(3) From relations of classes. Relation of S and P 
in Predication is twofold only either (a) a relation of 
identity or coincidence (of denotation or extension) or 
(b) a relation of denotational exclusion, while the relations 
possible between two classes are five. 

(4) From the relation of successive similar percepts 
to a conception or general notion which is implicated in 
every general name. When Mill says that the import 
of such propositions as : The colour I saw yesterday was 
a white colour, The sensation I feel is one of tightness, 
is to assert resemblance, he seems to confuse (1) with (4). 
If his account of the import of these two propositions is 
correct, then every proposition which has a general name 
for Predicate is a proposition " asserting" Resemblance, 
e.g. Rosa is fair-haired, This orange is ripe, Arsenic is 
a cause of death. 


(5) From the relation of members of a class (a) to 
each other or (b) to the class. 

Identical. Used to mean denotationally the same, 
the same individual or thing. 

System. By System is meant a group of two or more 
related objects or items. 

Logical Inference. When we can say that 

If a proposition (or pair of propositions) A is true, 

another proposition C is true ; 
then C is a logical inference (eductive or deduc 
tive) from A that is, the truth of C is implied in the 
truth of A, we cannot affirm C and deny A, C follows 
from A. 

Logical Inference has to be distinguished from 
Instinctive or " Psychological " Inference, and from what 
may be called Tentative Inference, which may be (1) a 
sudden apergu, a revelation, an intuition, or (2) an 
Hypothesis or guess, deliberately framed for purposes of 
investigation. " If we take the simplest possible case 
of mediate inference or syllogism (Deduction) we have 
S is M, M is P, entitling us to the inference S is P. 
Here we have, no doubt, as the conclusion, an assertion 
or proposition, which (qua assertion or proposition) differs 
in some way from either of the premisses ; or from both 
taken together. At the same time the content of the 
assertion S is P is certainly in some way contained in 
and justified by M is P and S is M. The exact connexion 
seems to me to be as follows : When as audience or seeker 
or pupil, we have learnt that M is P and S is M, and 
grasped the contents of the two assertions and their 
connexion, we find that we have really produced a con 
struction in which the connotations or intensions S and M 


and P are referred to one denotation (s,M,PJ. Having this 

whole before us as an object of thought, or imagination, 
it is apparent that it entitles us to say not only that 
S is M and M is P, but also that S is P, and even further, 
if we wish, that P is M and M is S and P is S, Not-P 
is not M, not-M is not S, etc. The same might be shown 
in a similar though even simpler way, of any case of 
Immediate Inference (Eduction). 

We may, of course, syllogise and otherwise infer in 
an unintelligent mechanical way, using the accepted laws 
of Formal Inference and of the Systems concerned as 
mere rules of thumb/ guides blindly obeyed. But if 
the acceptance of these rules can be justified, it must be 
seen that they are valid. It is, e.g., the vision actual or 
possible of a constructed whole, S that is M, M that is P, 

fs.M.Pj that justifies to our mind the assertion of (among 

other statements) S is P as an inference from 8 is M and 
M is P. 

I believe that a clear distinction between the contrasted 
attitudes of hearer, reader, or learner on the one hand, and 
speaker or teacher on the other hand, is very important 
indeed for the theory of Logical Inference." (See Mind, 
1908, pp. 533, 534.) 


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