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Full text of "Studies in logic"

PRESENTED 

TO 

THE UNIVERSITY OF TORONTO 



THE JOHNS HOP.KINS UNIVERSITY 



BALTIMORE 
1890 



STUDIES IN LOGIC, 



, 



BY MEMBERS 



OF THE 



JOHNS HOPKINS UNIVEKSITY. 



^J 



BOSTON: 
LITTLE, BROWN, AND COMPANY. 

1883. 



Copyright, 1883, 
BY C. S. PEIRCE. 




UNIVERSITY PRESS: 
JOHN WILSON AND SON, CAMBRIDGE. 



PKEFACE. 



THESE papers, the work of my students, have 
been so instructive to me, that I have asked 
and obtained permission to publish them in one 
volume. 

Two of them, the contributions of Miss Ladd 
(now Mrs. Fabian Franklin) and of Mr. Mitchell, 
present new developments of the logical algebra 
of Boole. Miss Ladd s article may serve, for 
those who are unacquainted with Boole s " Laws 
of Thought," as an introduction to the most won 
derful and fecund discovery of modern lo^ic. 
The followers of Bo#le have altered their mas 
ter s notation mainly in three respects. 

1. A series of writers, Jevons, in 1864; 
Peirce, in 1867; Grassman, in 1872; Schroder, 
in 1877 ; and McColl in 1877, successively and 
independently declared in favor of using the sign 
of addition to unite different terms into one aggre 
gate, whether they be mutually exclusive or not. 
Thus, we now write 

European + Eepublican, 

to stand for all Europeans and Republicans taken 



IV PREFACE. 

together, without intending to count twice over 
the European Republicans. Boole and Venn (his 
sole living defender) would insist upon our writ 
ing 

European + Non-European Bepublican, 

or 

Non-Republican European + Bepublican. 

The two new authors both side with the ma 
jority in this respect. 

2. Mr. McColl and I find it to be absolutely 
necessary to add some new sign to express exist 
ence ; for Boole s notation is only capable of 
representing that some description of thing does 
not exist, and cannot say that anything does exist. 
Besides that, the sign of equality, used by Boole 
in the desire to assimilate the algebra of logic to 
that of number, really expresses, as De Morgan 
showed forty years ago, a complex relation. To 
say that 

African Negro 

implies two things, that every African is a Negro, 
and that every Negro is an African. For these 
reasons, Mr. McColl and I make use of signs of 
inclusion and of non-inclusion. Thus, I write 

Griffin -< breathing fire 

to mean that every griffin (if there be such a 
creature) breathes fire; that is, no griffin not 
breathing fire exists; and I write 

Animal -< Aquatic, 

to mean that some animals are not aquatic, or 



PREFACE. V 

that a non-aquatic animal does exist. Mr. McColl s 
notation is not essentially different. 

Miss Ladd and Mr. Mitchell also use two signs 
expressive of simple relations involving existence 
and non-existence; but in their choice of these 
relations they diverge both from McColl and me, 
and from one another. In fact, of the eight sim 
ple relations of terms signalized by De Morgan, 
Mr. McColl and I have chosen two, Miss Ladd 
two others, Mr. Mitchell a fifth and sixth. The 
logical world is thus in a situation to weigh the 
advantages and disadvantages of the different 
systems. 

3. The third important modification of Boole s 
original notation consists in the introduction of 
new signs, so as to adapt it to the expression of 
relative terms. This branch of logic which has 
been studied by Leslie Ellis, De Morgan, Jo 
seph John Murphy, Alexander MacFarlane, and 
myself, presents a rich and new field for investi 
gation. A part of Mr. Mitchell s paper touches 
this subject in an exceedingly interesting way. 

The method of using the Boolian calculus 
already greatly simplified by Schroder and by 
McColl receives still further improvements at 
the hands both of Miss Ladd and Mr. Mitchell, 
and it is surprising to see with what facility their 
methods yield solutions of problems more intri 
cate and difficult than any that have hitherto been 
proposed. 



VI PEEFACE. 

The volume contains two other papers relating 
to deductive logic. In one of these Mr. Grilman 
develops those rules for the combination of rela 
tive numbers of which the general principles of 
probabilities are special cases. In the other, Dr. 
Marquand shows how a counting machine, on a 
binary system of numeration, will exhibit De 
Morgan s eight modes of universal syllogism. 

There are, besides, two papers upon inductive 
logic. In the first, Dr. Marquand explains the 
deeply interesting views of the Epicureans, known 
to us mainly through the work of Philodemus, 
irepl <T77/xeiW KOLI crrjjjLeictxTewv, which exists in a 
fragmentary state in a Herculaneum papyrus. 

The other paper is one which, at the desire of 
my students, I have contributed to the collection. 
It contains a statement of what appears to me to 
be the true theory of the inductive process, and 
the correct maxims for the performance of it. 
I hope that the thoughts that a long study has 
suggested to me may be found not altogether 
useless to those who occupy themselves with the 
application of this kind of reasoning. 

I have to thank the Trustees of the Johns 
Hopkins University, for a very liberal contribu 
tion toward the expenses of this publication. 

C. S. PEIKCE. 
BALTIMORE, Dec. 12, 1882. 



CONTENTS. 



PAGB 

THE LOGIC OP THE EPICUREANS 1 

By Allan Marguand. 



A MACHINE FOR PRODUCING SYLLOGISTIC VARIATIONS 12 

By Allan Marquand. 

NOTE ON AN EIGHT-TERM LOGICAL MACHINE ... 16 
ON THE ALGEBRA OF LOGIC 17 

By Christine Ladd. 

ON A NEW ALGEBRA OF LOGIC 72 

By 0. H. Mitchell. 

OPERATIONS IN EELATIVE NUMBER WITH APPLICA 
TIONS TO THE THEORY OF PROBABILITIES . . . 107 
By B. I. Oilman. 

A THEORY OF PROBABLE INFERENCE 126 

NOTE A 182 

NOTE B 187 

By C. S. Peirce. 



THE LOGIC OF THE EPICUREANS. 

BY ALLAN MARQUAND. 



WHEN we think of the Epicureans we picture a friendly 
brotherhood in a garden, soothing eacli other s fears, and 
seeking to realize a life of undisturbed peace and happi 
ness. It was easy, and to their opponents it became 
natural, to suppose that the Epicureans did not concern 
themselves with logic ; and if we expect to find in their 
writings a highly developed formal logic, as that of the 
Stoics, our search will be in vain. But if we examine 
the letters of Epicurus, the poem of Lucretius, and the 
treatise of Philodemus 1 with a view to discovering the 
Epicurean mode of thought, we find a logic which out 
weighs in value that of their Stoic rivals. This logic is 
interesting to us, not only because it is the key to that 
school of Greek Philosophy which outlasted every other, 
but because a similar logic controls a powerful school 
of English thought. 

The logic of Epicurus, like that of J. S. Mill, in op 
position to conceptualism, attempts to place philosophy 
upon an empirical basis. Words with Epicurus are signs 
of things, and not, as with the Stoics, of our ideas of 

1 Gomperz: Herkulanische Studien I. Leipzig, 1865. Bahnsch: Des 
Epicureers Philodemus Schrift Ilepi o-wduv nai ffweidxreuv. Eine Darleg- 
ung ihres Gedankengehalts. Lyck, 1879. 

1 



2 THE LOGIC OF THE EPICUKEANS. 

things. 1 There are, therefore, two methods of inquiry : 
One seeks for the meanings of words ; the other, for a 
knowledge of things. The former is regarded as a pre 
liminary process ; the latter, the only true arid necessary 
way of reaching a philosophy of the universe. 

All our knowledge is to be brought to the test of 
sensation, pre-notion, and feeling. 2 By these we do not 
understand three ultimate sources of knowledge. De- 
mocritus 3 held to only one source, viz., Feeling ; and Epi 
curus, who inherited his system, implicitly does the same. 
But each of these modes of feeling has its distinguishing 
characteristic, and may be used to test the validity of our 
knowledge. It is the peculiarity of sensation to reveal to 
us the external world. Sensation 4 reasons not, remem 
bers not ; it adds nothing, it subtracts nothing. What 
it gives is a simple, self-evident, and true account of 
the external world. Its testimony is beyond criticism. 
Error arises after the data of sensation become involved 
in the operations of intellect. If we should compare this 
first test of truth with Hume s " impressions," the second 
test, pre-notion, would correspond with Hume s " ideas." 
Pre-notions 5 were copies of sensations in a generalized or 
typical form, arising from a repetition of similar sensa 
tions. Thus the belief in the gods 6 was referred to the 
clear pre-notions of them. Single effluxes from such re 
fined beings could have no effect upon the senses, but 
repeated effluxes from deities sufficiently similar produce 
in our minds the general notion of a god. 7 In the same 

1 The hypothesis of XeKrd, or of immaterial notions, was a conceptu- 
alistic inconsistency on the part of the Stoics. The Epicureans and the 
more consistent empiricists among the Stoics rejected them. -Sextus 
Empiricus, Math. viii. 258. 

2 Diogenes Laertius, x. 31. 8 Sextus: Math., vii. 140. 
* D. L., x. 31. 6 D. L., x. 33. 6 D. L., x. 123, 124. 

Cicero: De Nat. Deor., i. 49; D. L,. x. 139. 



THE LOGIC OF THE EPICUREANS. 3 

manner, but through the senses, the continued observa 
tion of horses or oxen produce in us general notions, to 
which we may refer a doubt concerning the nature of the 
animal that moves before us. 

The third criterion, Feeling (in the limited sense), was 
the ultimate test for ethical maxims. The elementary 
forms are the feeling of pleasure and the feeling of pain. 
A fourth criterion was added, viz., The Imaginative rep 
resentations of the intellect. Its use is by no means 
clear. 

Upon this foundation rises the structure of Epicurean 
logic. When we leave the clear evidence of sense we 
pass into the region of opinion, away from the stronghold 
of truth to the region where error is ever struggling for 
the mastery of our minds. A true opinion l is character 
ized as one for which there is evidence in favor or none 
against ; a false opinion, one for which there is no evi 
dence in favor or some against. The processes by which 
we pass to the more general and complex forms of know 
ledge are four : Observation, Analogy, Resemblance, Syn 
thesis. 2 By Observation, we come into contact with the 
data of the senses ; by Analogy, we may not only enlarge 
and diminish our perceptions, as we do in conceiving a 
Cyclops or a Pygmy, but also extend to the unperceived 
the attributes of our perceptions, as we do in assigning 
properties to atoms, the soul, and the gods ; by Resem 
blance, we know the appearance of Socrates from having 
seen his statue ; by Synthesis, we combine sensations, as 
when we conceive of a Centaur. 

As a matter of fact, Epicurus regards only two proces 
ses, Observation and Analogy. Our knowledge, then, 

1 D. L., x. 34, 51. Sextus: Math., vii. 211. 

2 D. L., x. 32. The Stoics held a similar view; see D. L., vii. 52. 



4 THE LOGIC OF THE EPICUREANS. 

consists of two parts : l (1) The observed, or phenomena 
clear and distinct to consciousness ; and (2) The unob 
served, 2 consisting of phenomena which are yet to be ob 
served, and of hidden causes which lie forever beyond 
our observation. The function of logic 3 consists in in 
ference from the observed to the unobserved. This was 
called a sien-iiiference. According to Epicurus there are 
two methods 4 of making such an inference; one resulting 
in a single explanation, the other in many explanations. 
The former may be illustrated by the argument, Motion 
is a sign of a void. Here the void is regarded as the 
only explanation to be given of motion. In other cases 
many explanations are found equally in harmony with 
our experience. All celestial phenomena belong to this 
class. That explanation which alone represents the true 
cause of such a phenomenon being unknown, we must 
be content to admit many explanations as equally prob 
able. Thus thunder 5 is explained by supposing either 
that winds are whirling in the cavities of the clouds, or 
that some great fire is crackling as it is fanned by the 
winds, or that the clouds are being torn asunder or are 
rubbing against each other as they become crystallized. 
In thus connecting celestial and terrestrial phenomena, 
Epicurus aimed only to exclude supernaturalistic expla 
nations. This done, he was satisfied. 

In the garden at Athens this logic took root and grew ; 
and by the time that Cicero visited Greece and sat at the 
feet of Zeno, 6 he may have listened to that great repre- 

1 Philodemus: Rhet,, lib. iv., i. col. xix. 

2 That is, TO Trpocr^vov Kai TO &St]\ov, D. L., x. 38. 

8 D. L., x. 32. 66 ev /ecu irepl TWV ddyXuv diro T&V <f>au>0(J,frwv Xp?j <T77/xei- 
ovcrOai. 

* Ibid., x. 86, 87. 

5 Ibid., x. 100. Cf. Lucretius, lib. vi. 95-158. 

6 See Zeller s Stoics, Epicureans, and Sceptics. London, 1880, p. 412, 
n. 3. 



THE LOGIC OF THE EPICUKEANS. 5 

sentative of the Epicurean School discussing such ques 
tions 1 as, How may we pass from the known to the 
unknown ? Must we examine every instance before 
we make an induction ? Must the phenomenon taken 
as a sign be identical with the thing signified ? Or, if 
differences be admitted, upon what grounds may an in 
ductive inference be made ? And, Are we not always 
liable to be thwarted by the existence of exceptional 
cases ? But such questions had no interest for Cicero. 
He was too much an orator and rhetorician to recognize 
the force of the . Epicurean opposition to dialectic. The 
Epicurean logic 2 to him was barren and empty. It made 
little of definition ; it said nothing of division ; it erected 
no syllogistic forms ; it did not direct us how to solve 
fallacies and detect ambiguities. And how many have 
been the historians of philosophy who have assigned 
almost a blank page to Epicurean logic ! 

With a supreme confidence in the truth of sensation 
and the validity of induction the Epicureans stood in con 
flict with the other schools of Greek philosophy. The 
Stoics, treating all affirmation from the standpoint of the 
hypothetical proposition, acknowledged the validity of 
such inductions only as could be submitted to the modus 
tottens. The Sceptics denied the validity of induction 
altogether. Induction was treated as a sign-inference, 
and a controversy appears to have arisen concerning 
the nature of signs, as well as concerning the mode 
and validity of the inference. The Stoics divided signs 
into suggestive and indicative? By means of a sug 
gestive sign we recall some previously associated fact : 
as from smoke we infer fire. By indicative signs we 
infer something otherwise unknown : thus motions of 

1 Philodemus TTC/H <yrnj.etC)v, col. xix.-xx. 2 Cicero: De Fin., i. 7, 22. 
3 See PraiuUl s Ges. d. Log., i. 458. 



6 THE LOGIC OF THE EPICUKEANS. 

,the body are signs of the soul. Objectively a sign was 
viewed as the antecedent of a valid conditional propo 
sition, implying a consequent. Subjectively, it was a 
thought, mediating in some way between things on the 
one hand, and names and propositions on the other. 
The. Epicureans looked upon a sign as a phenomenon^ 
from whose characters we might infer the characters of 
other phenomena under conditions of existence suf 
ficiently similar. The sign was to them an object of 
sense. In considering the variety of signs, the Epicureans 
appear to have admitted three kinds ; but only two are 
defined in the treatise of Philodemus. 1 A general sign is 
described as a phenomenon which can exist whether the 
thing signified exists or not, or has a particular character 
or not. A particular sign is a phenomenon which can 
exist only on the condition that the thing signified act 
ually exists. The relation between sign and thing sig 
nified in the former case is resemblance ; in the latter, it 
is invariable sequence or causality. The Stoics, in devel 
oping the sign-inference, inquired, How may we pass from 
the antecedent to the consequent of a conditional prop 
osition ? They replied, A true sign exists only when 
both antecedent and consequent are true. 2 As a test, 
we should be able to contrapose the proposition, and see 
that from the negative of the consequent the negative of 
the antecedent followed. Only those propositions which 
admitted of contraposition were allowed to be treated as 
hypothetical. 3 

On this prepositional ground, therefore, the Epicurean 
must meet his opponent. This he does by observing 
that general propositions are obtained neither by contra 
position nor by syllogism, nor in any other way than 

1 Philod., loc. cit., col. xiv. 2 Sextus : Math., viii. 256. 
3 Cicero: De Fato, 6, 12; 8, 15. 



THE LOGIC OF THE EPICUKEANS. 7 

by induction. 1 The contraposed forms, being general 
propositions, rest also on induction. Hence, if the in 
ductive mode of reasoning be uncertain, the same degree 
of uncertainty attaches to propositions in the contra- 
posed form. 2 The Stoics, therefore, in neglecting in 
duction, were accused of surrendering the vouchers by 
which alone their generalizations could be established. 3 
In like manner they were accused of hasty generalization, 
of inaccurate reasoning, of adopting myths, of being rhet 
oricians rather than investigators of Nature. Into the 
truth of these accusations we need not inquire. It is 
enough that they cleared the way for the Epicureans to 
set up a theory of induction. 

The first question which Zeno sought to answer was, 
" Is it necessary that we should examine every case of 
a phenomenon, or only a certain number of cases ? " 4 
Stoics and Sceptics answered, The former is impossi 
ble, and the latter leaves induction insecure. But Zeno 
replied : " It is neither necessary to take into considera 
tion every phenomenon in our experience, nor a few cases 
at random ; but taking many and various phenomena of 
the same general kind, and having obtained, both from 
our observation and that of others, the properties that are 
common to each individual, from these cases may we 
pass to the rest." 5 Instances taken from a class and 
exhibiting some invariable properties are made the basis 
of the inductive inference. A certain amount of variation 
in the properties is not excluded. Thus from the fact that 
the men in our region of country are short-lived, we may 
not infer that the inhabitants of Mt. Athos are short 
lived also ; for " men in our experience are seen to vary 
considerably in respect to length or brevity of life." 

1 Philod., loc. cit., col. xvii. 2 Ibid., col. ix. 

3 Ibid., col. xxix. * Ibid., col. xix. 13-15. 

5 Ibid., col. xx. 30-col. xxi. 3. 6 Ibid., col. xvii. 18-22. 



THE LOGIC OF THE EPICUREANS. 

Within limits, then, we may allow for variation due to the 
influence of .climate, food, and other physical conditions ; 
but our inference should not greatly exceed the limits of 
our experience. But, in spite of variations, there are 
properties which in our experience are universal. Men 
are found to be liable to disease and old age and death ; 
they die when their heads are cut off, or their hearts 
extracted; they cannot pass through solid bodies. By 
induction we infer that these characteristics belong to 
men wherever they may be found, and it is absurd to 
speak of men under similar conditions as not susceptible 
to disease or death, or as having the ability to pass through 
iron as we pass through the air. 1 

The Epicurean looks out upon Nature as already di 
vided and subdivided into classes, each class being closely 
related to other classes. The inductive inference proceeds 
from class to class, not in a hap-hazard way, but from one 
class to that which resembles it most closely. 2 In case the 
classes are identical, there is no distinction of known and 
unknown ; and hence, properly speaking, no inductive in 
ference. 3 In case the classes are widely different, the 
inference is insecure. But within a certain range of re 
semblance we may rely as confidently upon an inductive 
inference as we do upon the evidence of sense. 4 

In speaking of the common or essential characters, the 
basis of induction, it was usual to connect them with the 
subject of discourse by the words ? ;, K a06, or nrapo. These 
words may be taken in four senses : 5 (1) The properties 
may be regarded as necessary consequences ; so we may 
say of a man that he is necessarily corporeal and liable 
to disease and death. (2) Or as essential to the concep 
tion or definition of the subject. This is what is con- 

1 Philod., loc. cit., col. xxi. 

2 Ibid., col. xviii. 20 ; col. xxviii. 25-29. 3 Ibid., col. vi. 8-10. 

4 Ibid., Frag. 2, 5-6. 6 Ibid>j col> xxxiii> 33 _ col> xxxiv 34 



THE LOGIC OF THE EPICUREANS. 9 

veyed in the expression, " Body as body has weight and 
resistance ; man as man is a rational animal." (3) That 
certain properties are always concomitant. (4) The 
fourth sense, lost in the lacunas, appears from the fol 
lowing examples to involve degree or proportion : " The 
sword cuts as it has been sharpened ; atoms are im 
perishable in so far as they are perfect ; bodies gravitate 
in proportion to their weight." 

Zeno s theory of induction may be formulated in the 
following Canons : 

CANON I. If we examine many and various instances 
of a phenomenon, and find some character common to 
them all, and no instance appears to the contrary, this 
character may be transferred to other unexamined in 
dividuals of the same class, and even to other closely 
related classes. 

CANON II. If in our experience a given character is 
found to vary, a corresponding amount of variation may 
be inferred to exist beyond our experience. 

The most important objection made to this theory was, 
that phenomena exist in our experience exhibiting pecu 
liar and exceptional characters, and that other exceptions 
might exist beyond our experience to vitiate any induc 
tion we may make. The following examples are given : l 
The loadstone has the peculiar property of attracting iron ; 
amber, of attracting bran ; the square number 4 X 4, of 
having its perimeter equal to its area. Exceptional char 
acters are found in the Alexandrian anvil-headed dwarf, 
the Epidaurian hermaphrodite, the Cretan giant, the 
pygmies in Achoris. The sun and moon also are unique ; 
so are time and the soul. Admitting such exceptional 
phenomena, the Epicurean replies, that the belief that a 
similar state of things exists beyond our experience can 

1 Philod., loc. cit., col. i., ii. 



10 THE LOGIC OF THE EPICUREANS. 

be justified only inductively. 1 And exceptional phenom 
ena must be viewed not as closely resembling, but as 
being widely different from, other phenomena. Induc 
tions concerning loadstones must be confined to load 
stones, and not extended to other kinds of stones. Each 
class of exceptional phenomena offered a new field for 
induction, and hence could be said to strengthen and not 
to weaken the inductive argument. 2 

The correctness of all inductions could be tested by 
the rule of Epicurus for the truth of opinion in general. 
An induction is true, when all known instances are in its 
favor, or none against; it is false, when no instances are 
in its favor, or some against. When the instances are 
partly one way and partly another, we cannot reach 
universal conclusions, but only such as are probable. 3 

This theory of induction was completed by a considera 
tion of fallacies, summarized in a work called the " Deme- 
triac." 4 These consisted in 

1. Failing to see in what cases contraposition is ap 
plicable. 

2. Failing to see that we should make inductions not 
in a hap-hazard way, but from properties which resemble 
each other very closely. 

3. Failing to see that exceptional phenomena are in 
no way at variance with the inductive inference, but on 
the other hand add to its force. 

4. Failing to observe that we infer from the known to 
the unknown, only when all the evidence is in favor and 
no shadow of evidence appears to the contrary. 

5. The failure to perceive that general propositions 
are derived not by contraposition, but by induction. 

When we compare the work of Zeno with that of 

1 Philod., loc. cit., col. xxv. 2 Ibid., col. xxiv. 10-col. xxv. 2. 
8 Ibid., col. xxv. 31-34. 4 Ibid., col. xxviii. 13-col. xxix. 24. 



THE LOGIC OF THE EPICUREANS. 11 

Epicurus, an important logical difference is brought to 
view. Both are occupied with the sign-inference, and 
look upon inference as proceeding from the known to the 
unknown. Epicurus, however, sought only by means of 
hypothesis to explain special phenomena of Nature. Zeno 
investigated generalizations from experience, with a view 
to discovering the validity of extending them beyond our 
experience. This resulted in a theory of induction, which, 
so far as we know, Epicurus did not possess. In the 
system of Aristotle, induction was viewed through the 
forms of syllogism, and its empirical foundation was not 
held in view. The Epicureans, therefore, were as much 
opposed to the Aristotelian induction, as they were to the 
Aristotelian syllogism. It was Zeno the Epicurean who 
made the first attempt to justify the validity of induction. 
The record of this attempt will give the treatise of Philo- 
demus a permanent value in the history of inductive 
logic. 

It is refreshing to see the formalistic and rhetorical 

o 

atmosphere which had surrounded the subject of logic 
breaking away, and an honest attempt being made to 
justify the premises of syllogism. As yet, this had not 
been done by all the moods of the philosophers. 

It is also interesting to find in the ancient world a 
theory of induction which rests upon observation, sug 
gests experiment, assumes the uniformity of Nature, and 
allows for the variation of characters. 



A MACHINE FOR PRODUCING SYLLOGISTIC 
VARIATIONS. 

BY ALLAN MARQUAND. 



FROM any syllogism a number of logical variations 
may be derived. One operation by which this may be 
accomplished is contraposition. This operation consists 
in effecting a change in the order of the terms of a 
proposition, the state of things which the proposition is 
designed to express being supposed to remain unchanged. 
Thus the state of things expressed by the proposition 
" every A is a B " may be expressed also by " every 
non-B is a non-A," or by the form, " there is a B for 
every A." 

We proceed now to apply this principle to the syllo 
gism. For our notation let us take letters A, B, C, 
etc. for general terms, and express their negatives by 
writing dashes over them, A, B, C, etc. Let a short 
curved mark over a letter indicate that its logical quan 
tity has been changed ; thus, A, B, C, etc. A general 
term will be thus made particular, and a term already 
particular will be made general. Let us use the sign 
-< for the copula. 1 We may then express the syllo 
gism Barbara in the form 

A-<B 

B-< C 

.-. A -< C 

1 This notation is that used by Mr. C. S. Peirce, "On the Logic of 
Relatives." Memoirs Am. Acad. of Arts and Sciences, vol. ix, 1870. 



A MACHINE FOR SYLLOGISTIC VARIATIONS. 13 

From this as a starting-point we may produce formal 
variations by various modes of contraposition. The ex 
hibition of two such forms will suffice. 

(1) We may regard the logical quality of the terms 
and contrapose. The form A -< B then becomes 
B-< A, or, "every non-B is a non-A." 

(2) We may regard the logical quantity of the terms 
and contrapose. The form A -< B then becomes 
B -< A. The latter form we may take to mean, " there 
is a B for every A," or " the B s include all the A s." 

Applying these two kinds of contraposition to Barbara, 
we obtain the following variations : 

Qualitative Variations. 

B<A A-<B B-<A A<B B-<A A-<B B-<A 
Fundamental B^CC 0<B 0-<B B-<C B<C CKB C-<B 

Form /. A-<C . . A<C .-.A-<C . .C-<A .-.C<A .-. C -< A .-. C -< A 

J Quantitative Variations. 

B-<C 



B-<A AKB B^A A-<B 
B-<0 C-<B C<B B^C 

.-. C-<A /. 



These may be classed as two figures according as the 
conclusion has the fundamental or contraposed form ; or 
they may be classed as four figures according as one or 
other, or both, or neither premise has been contraposed ; 
or as eight figures, if we regard merely the relative posi 
tion of the terms. The number of such variations may 
be indefinitely increased by admitting other modes of 
contraposition, or by starting from other syllogistic 
forms. All these variations may be easily produced 
by a mechanical contrivance. In order to secure this 
I have constructed a machine (Fig. 1) which pre 
sents to view three flaps in which are inserted cards 
containing the premises and conclusion of the syllogism 
which is to undergo transformation. Each flap, on 



14 A MACHINE FOR SYLLOGISTIC VARIATIONS. 



making a half-revolution, presents its proposition in a 
contraposed form. The flaps terminate on one side of 



FIG. l. 




Scale i in. 

the machine in one-inch brass friction wheels. These 
are marked a, 5, and c in Fig. 2. The wheels d, e, 
and / are, respectively, one, two, and four inches in 
diameter. Upon each of these wheels is fitted the sec 
tor of a wheel of like dimensions. Wheel d has on its 
outer side a sector of 180 ; wheel e, on its inner side, 
one of 90 ; wheel /, on its outer side, one of 45. The 
friction of these sectors against the wheels a, 5, and c 
causes the half-revolutions of the three flaps. By turn- 



A MACHINE FOR SYLLOGISTIC VARIATIONS. 15 



FIG. 2. 



ing a crank attached to wheel d, the proposition A -< B 
is contraposed at the end of every turn, B -< C at every 
alternate turn, and A -< C at the end of every fourth 
turn. Eight turns of 
the crank will exhibit 
seven variations, and 
restore the fundamen 
tal syllogism to view. 
This mechanism 
could be readily ex 
tended so as to pro 
duce variations in a 
Sorites. A Sorites of 
n propositions would 
require, to contrapose 
its conclusion, a wheel 
of 2 n ~ 1 inches in di 
ameter. We should 
secure, as in the syl 
logism, 2 n 1 varia 
tions for each kind 
of contraposition. 

Scale J in. 

NOTE. The Syllogistic Variation Machine will unfold to view 
the combinations of three logical terms and their negatives ; or if we 
take the letters B C, A U, D T, we obtain the words 

BCBCBCBC 

AAUUAAUU 
DDDDTTTT 




NOTE ON AN EIGHT-TERM LOGICAL MACHINE. 



I HAVE completed the design of an 8-term Logical 
Machine, of which a 4-term model is now nearly fin 
ished. If the premises be reduced to the form of the 
combinations to be excluded, as suggested by Boole and 
carried out by Venn, the operation of excluding these 
combinations may be performed mechanically by this 
machine. I have followed Jevons in making use of keys, 
but require for the 8-term machine only eight positive 
and eight negative letter keys and two operation keys. 
The excluded combinations are exhibited by indicators, 
which fall in the squares of one of my logical diagrams 
(Phil. Mag. ON. 81) from the perpendicular to a hori 
zontal position. The non-excluded combinations, which 
constitute the conclusion, are exhibited by the indicators 
which are left standing. 



ON THE ALGEBRA OF LOGIC. 

BY CHRISTINE LADD. 



THERE are in existence five algebras of logic, those 
of Boole, Jevons, Schroder, McColl, and Peirce, of 
which the later ones are all modifications, more or less 
slight, of that of Boole. I propose to add one more to 
the number. It will bear more resemblance to that of 
Schroder than to any of the others; but it will differ 
from that in making use of a copula, and also in the 
form of expressing the conclusion. 1 

ON IDENTICAL PROPOSITIONS. 

The propositions which logic considers are of two 
kinds, those which affirm the identity of subject and 
predicate, and those which do not. Algebras of logic 
may be classified according to the way in which they 
express propositions that are not identities. Identical 
propositions have the same expression in all. Of the 
logical theorems which are identities, I shall give those 
which are essential to the subject, and for the most part 
without proof. 

(1) The sign is the sign of equality, a = 5, a 
equals #, means that in any logical expression a can 

1 The substance of this paper was read at a meeting of the Metaphysical 
Club of the Johns Hopkins University, held in January, 1881. 



18 ON THE ALGEBRA OF LOGIC. 

be substituted for 6, or b for #, without change of value. 
It is equivalent to the two propositions, " there is no a 
which is not 5," and, " there is no b which is not a." 

(2) The negative of a term or a proposition or a 
symbol is indicated by a line drawn over it. a = what 
is not a. 



(3) a + b = what is either 
a or b. As a class, it takes in 
the whole of a together with 
the whole of b, what is com 
mon to both being counted 
once only. It has the quality 
of either a or b, and hence 
the quality of the entire class 
is the quality common to a 
and b. The only qualities pos- 



(3 ) aXb = what is both a 
and b. As a class, it is what 
is common to the classes a and 
b. As a quality, it is the 
combination of all the quali 
ties of a with all the qualities 
of b. When relative terms 
(XXI) 1 are excluded from 
consideration, ab may be writ 
ten for a X b. 

sessed by every member of the 
class " lawyers and bankers * are the qualities which lawyers 
-and bankers have in common. 

When arithmetical multiplication and addition are to 
be considered at the same time, logical multiplication 
and addition may be indicated by enclosing + and X in 
circles. The addition of logic has small connection 
with the addition of mathematics, and the multiplication 
has no connection at all with the process whose name it 
has taken. The object in borrowing the words and the 
signs is to utilize the familiarity which one has already 
acquired with processes which obey somewhat similar 
laws. There would not be the slightest difficulty in 
inverting the operations, and expressing logical multi 
plication in terms of addition, and logical addition in 
terms of multiplication. The essential processes of sym 
bolic logic are either addition or multiplication (for 
greater convenience, both are used), and negation. The 
1 References in Roman numerals are to the titles at the end. 



ON THE ALGEBRA OF LOGIC. 19 

latter process renders any inverse processes which might 
correspond to subtraction and division quite unneces 
sary, and it is only on account of a supposed resemb 
lance between the logical and the mathematical processes 
that an attempt to introduce them has been made. 



(4 ; ) aaa = a. 

(5 r ) abc = bca = cba. 
(6 ) 



(4) a + a + . . . = a. 

(5) a+b+c=b+c+a=c+b+a. 

(6) a + be=(a + b)(a + c). 
The symbol oo represents the universe of discourse. 
(Wundt, Peirce.) It may be the universe of conceivable 
things, or of actual things, or any limited portion of 
either. It may include non-Euclidian w-dimensional 
space, or it may be limited to the surface of the earth, 
or to the field of a microscope. It may exclude things 
and be restricted to qualities, or it may be made co 
extensive with fictions of any kind. In any proposition 
of formal logic, oo represents wha,t is logically possible ; 
in a material proposition it represents what exists. 
(Peirce.) The symbol is the negative of the sym 
bol oo ; it denotes either what is logically impossible, or 
what is non-existent in an actual universe of any degree 
of limitation. 



(7 ) 
(80 a = 

(9 )ao=a+oo = a+(H5)+... 
(100 ab + al) + ab + aB 



(7) a + a = oo. 

(8) a=a+0 = a+bt+cc+ 

(9) Q = aQ = 

(10) 



The first member of this equa 
tion is called the complete development of two terms. The 
complete development of n terms, (a + a) (b + 1) (c + c) . . . , 
consists of the sum of 2" combinations of n terms each. 



(11 ) a+ab + abc+ ... = a \ (11) a(a+1>)(a+b+c). . .=a. 
This is called by Schroder the law of absorption. 



20 ON THE ALGEBRA OF LOGIC. 

The only process which presents any difficulty in this 
calculus is the process of getting the negative of a com 
plex expression ; and that difficulty is very slight if the 
right method is selected. There are three different 
methods, of which the last is of most frequent use. 
The first proceeds from the consideration that ab + al + 
db + ab is a complete universe (10 ), and that what is 
not one portion of a universe must be some other portion, 
if it exists at all. It follows that 

ab ah + ab + ab, 

(12) ab + al = ab + 5, 

ab + ab + ab = abj 

and the process is the same for the complete develop 
ment of any number of terms. This is the only rule 
made use of by Boole and by Mr. Jevons for obtaining a 
negative. If certain combinations of ten terms are 
given as excluded, to get those which are not excluded 
it is necessary, by this method, to examine 1,024 combi 
nations of ten terms each. 

The second method is contained in the following 
formulas : 



(13 ) 



ab a + b. 



(13) a + b = al. 
a + I = ab. 



That is, the negative of a product is the sum of the 
negatives of the terms, and the negative of a sum is the 
product of the negatives of the terms. 1 For example. 

1 Professor Wundt (XVIII., p. 257, note) makes the singular mistake 
of supposing that because x (y -\- 2) = xy -\- xz, the parentheses must be 
removed before performing any general operation upon an expression. 
The negative of a product of the form (a -f- & -f- <?) m, he says, is not 
dbc-{-m, but (a-f-m) (b-\-m) (c-}-m); and in working his problems he 
actually expresses it in this way, performs the indicated multiplication, 
obtaining doc -f- (a-f- 5 -f- c) m-\-m, and then reduces this expression by 
the absorption law (11 ) to aoc-f- ra. 



ON THE ALGEBRA OF LOGIC. 21 



a + be + def= a (5 + c) (a + e +/). 

This rule was first given by De Morgan (" On the 
Syllogism," No. III., 1858). It may be proved in the 
following way : 

by (12), 



a + b = a (b + b) + (a + a) b 
= ~ab~- 



ab ab + al> + aft + aB 
= a (b + 1) + (a + a) I 

by (12). 

It appears that with the use of the negative sign the 
sum and the product are not both essential to complete 
expression. A sum can be expressed as the negative of 
a product, or a product can be expressed as the negative 
of a sum. The dualism which has been pointed out by 
Schroder, and which he indicates by printing his theo 
rems in parallel columns, is, then, not an essential quality 
of things, but merely an accident of language. We prefer 
to say " what is either black or blue," to saying " what 
is not at the same time both not black and not blue ; " but 
one is as easy to express symbolically as the other. It 
would not be difficult to develop the whole subject in terms 
of multiplication alone, or of addition alone ; but the gain 
in simplicity is not equal to the loss in naturalness. 

The third method of obtaining the negative of an ex 
pression is by means of the following equation : 



(14) pab + qal + rob + sab pab + qab + fab + sab. 

That is, consider any number of the letters as the 
elements of a complete development (10 ), and take the 
negative of their coefficients. The reason is the same 
as for (12), the two expressions together make up a 
complete universe, since 

pab + pa b = ab, etc. 
It is necessary to observe that if any part of the develop- 



22 ON THE ALGEBRA OF LOGIC. 

ment is wanting, its coefficient is 0, and the negative of 
its coefficient is QO . For instance, 



O + q + r) xy + stxy + uvwxy 
=pqrxy + ( + t) xy + (u + v + w) xy + xy. 

The entire number of combinations excluded by the 
first member is 7.2 5 + 2 6 + 2 5 , and that included by 
the second member is 2 5 + 3.2 6 + 7.2 6 + 2 8 , and together 
they make up 1024. This rule is given by Schroder only 
(XIV., p. 19). It is much easier of application than 
(12) or (13), except when the given expression bears no 
resemblance to a complete development. 

(15) An expression may be said to be in its simplest 
form when it is represented by the smallest possible 
number of letters. It does not follow that it is then in 
its least redundant form. For instance, in 

a + b, = a + dbj = dL + b, 

a + b is simpler than either of the other expressions, 
but it is redundant. It is 

a (b + 1) + (a + a) b, 

which contains the combination ab twice ; while 
a + abj = a (b + I) + ab, 

contains each combination once only. The reduction 
of an expression to its simplest form may usually be 
accomplished by inspection. Take, for example, the 
expression 

a + be + abd + add. 
We have 

a+ a($ + c)d= a + bed, 
and 

be + bed = be + d. 

Hence the whole expression is 

a + be + d. 



ON THE ALGEBRA OF LOGIC. 23 

If the reduction is not evident, it may be facilitated by 
taking the negative of the expression, reducing it, and 
then restoring it to the positive form (XVI., vol. x. 
p. 18). 

OX THE COPULA. 

I shall adopt the convention by which particular pro 
positions are taken as implying the existence of their 
subjects, and universal propositions as not implying the 
existence of their subjects. Mr. Jevons would infer that 
the two propositions 

The sea-serpent is not found in the water, 
The sea-serpent is not found out of the water, 

are contradictory ; but Mr. McColl, Mr. Venn, and Mr. 
Peirce would infer that the sea-serpent does not exist. 
With this convention, contradiction can never exist 
between universal propositions nor between particular 
propositions taken by themselves. A universal propo 
sition can be contradicted only by a particular propo 
sition, and a particular only by a universal. The above 
premises are inconsistent with 

The sea-serpent has (at least once) been found. 

With this convention, hypothetical and categorical pro 
positions receive the same formal treatment. If , then 
b = all a is b a implies b. (Peirce.) 

Algebras of Logic may be divided into two classes, ac 
cording as they assign the expression of the " quantity " 
of propositions to the copula or to the subject. Algebras 
of the latter class have been developed with one copula 
only, the sign of equality ; for an algebra of the 
former class two copulas are necessary, 1 one universal 

1 Every algebra of logic requires two copulas, one to express propo 
sitions of non-existence, the other to express propositions of existence. 
This necessarily follows from Kant s discussion of the nature of the affir 
mation of existence in the " Critik der reinen Vernunft." 0. S. Peirce. 



24 



ON THE ALGEBEA OF LOGIC. 



and one particular. The following are the propositional 
forms which have been used by the principal recent 
Avriters on the algebra of logic : 1 





Traditional. 


Boole 
and 
Schroder. 


Jevons 
and 
Grassmann. 


Grassmann. 


McColl. 


Peirce. 


Uni 
versal. 


All a is b 

No a is b 


a=vb 
a=vl) 


a = ab 
a = aB 


a + b=b 
a +1=1 


a : b 

a : 1} 


a-<b 
a-<l 


Partic 
ular. 


Some a is b 
Some a is not & 


va = vb 
va = vfi 


ca = cab 
ca=cab 


ca-{-b = b 
ca-{-b=:b 


a-f-5 

a^-b 


a^<l 
a^<b 



v is a special symbol, used to denote an arbitrary, 
indefinite class. It is immediately eliminated from the 
universal propositions, which then appear in the forms 
ab = 0, ab = 0, respectively. In particular propositions 
" v is not quite arbitrary, and therefore must not be eli 
minated" (III., p. 124). Jevons makes no distinction 
between an indefinite class symbol, c, and any other 
class symbol. With Mr. McColl, every letter denotes a 
statement. By a : b is meant that the statement that 
any object is a implies the statement that it is also 5; 
but this does not affect the working of the algebra. 
The negative copula, -f-, is the denial of the affirmative 
copula, : , arid a -f- b , or, as he also writes it, (a : 5 ) , is 
read " a does not imply non-5." Mr. Peirce s symbol for 
the same copula is a modification of ^. a ^< b is the 
denial of a -< 6, and is read, " a is not wholly contained 
under 6." a and b may be either terms or propositions. 
The copula -< has an advantage over : in that it ex 
presses an unsymmetrical relation by an unsymmetrical 



1 Mr. Venn has collected some two dozen ways in which "a is b" has 
been put into logical form. 



ON THE ALGEBRA OF LOGIC. 25 

symbol, a -< b may be written b >- a and read, " b 
contains a." 

This quantified copula (-< or :) is positive for uni 
versal propositions, and negative for particular proposi 
tions. Another kind of quantified copula is possible, 
namely, one which is particular when positive, and uni 
versal when negative. Instead of writing 

( A^< B 

\ and \ A is-not-wholly B 
(or A is-partly-not B, 

we might write 

AVB I an( l J AV B 

A is-wliolly-not B ) (A is-partly B, 

and it will appear that this latter plan has certain advan 
tages. It comes perhaps a little nearer to common use. 
The sense " wholly " is usually attached to both is and 
is-not, but somewhat more strongly to the latter than to 
the former. We say, for instance, " flowers are fra 
grant," meaning that flowers are nearly always fra 
grant ; but " leaves are not blue " means that leaves 
are absolutely never blue. " Knives are sharp " would 
be taken as true ; " knives are not blunt " would excite 
opposition in the mind of the hearer. 

The sign V is a wedge, sign of exclusion. A V B is to 
be read " A is-not B," or " A is excluded from B" The 
sign V is an incompleted wedge, sign of incomplete ex 
clusion. A V B is to be read " A is in part B" or 
u A is not-wholly excluded from B" V is made into V 
by the addition of the negative sign ; what is not not 
wholly excluded from anything is wholly excluded from 
it. AvB and AvB are contradictory propositions ; 
each simply denies the other. 

The eight propositions of De Morgan are then, 



26 ON THE ALGEBRA OF LOGIC. 

A V B A is-not B ; no A is B. 

A V B A is in part B ; some A is .#. 

^ V B A is-not not- ; all A is ^. 

A V B A is partly not-1? ; some A is not .B. 

AVB What is not ^4 is-not B ; .4 includes all B. 

V ^ What is not A is in part B; A does not include all B. 

What is not A is-not not-^; there is nothing he- 
sides A and B. 

What is not ^ is in part not-B ; there is something 
besides A and B, 

where V connects terms that, exist, while V connects 
terms which may be non-existent. Only six of these 
propositions are distinct, since there is no difference of 
form ^between A\/S and A\/, nor between A V 5? 
and A V B. 

Propositions expressed with the copula : or -< are 
called inclusions ; propositions expressed with the cop 
ula V may be called exclusions. Exclusions with an 
even number of negative signs are positive (affirmative) 
propositions ; those with an odd number are negative 
propositions (De Morgan, " Syllabus of a Proposed Sys 
tem of Logic," p. 22). But the distinction, as Professor 
Wundt and others have pointed out, is unimportant. The 
only division of propositions which is of consequence is 
the division into universal and particular. The copulas 
V and V are intransitive copulas, a kind of copula of 
which De Morgan proposed to investigate the characters 
(" Syllabus," p. 31). They are symmetrical copulas, and 
the propositions A V B, A V B, may be read either for 
ward or backward. It is from the fact that there is no 
formal difference between subject and predicate that the 
advantages of this algebra follow. There is, however, a 
slight difference in meaning between A\/B and B V A ; 



ON THE ALGEBKA OF LOGIC. 27 

the subject of the proposition is more evidently the subject 
of discourse. The propositions, " no men are mortal/ 
and " there are no mortal men," convey the same infor 
mation; but the first offers it by way of information 
about men, and the second by way of a description of 
the universe. Information may be given about a pre 
dicate by the use of a different kind of copula; as in 
" no lack of hospitality is found among Baltimoreans." 

An inclusion is changed into the equivalent exclusion 
by changing the sign of the predicate. When an exclusion 
is to be made into an inclusion, it is a matter of indiffer 
ence which of its terms is regarded as predicate ; every 
exclusion contains within itself two inclusions, of which 
each is the converse by contraposition of the other. 
That is to say, 



AV B = A^<B = B^<^.. 

With this copula, therefore, the consideration of the con 
version of propositions is rendered unnecessary. So also 
is the consideration of the quantification of the predicate. 
With the copula -< the subject and predicate have un 
like quantity, or, more exactly, the quantity of the 
subject is universal and that of the predicate is indeter 
minate ; -< means either equal to or less than. But 
with the copula V the quantity of both subject and pre 
dicate is universal, and with its denial V both subject 
and predicate are taken in part only. 

The copula -< must be taken in an inverted sense 
according as subject and predicate are taken in exten 
sion or in intension ; but the copula V possesses the same 
meaning, whatever interpretation one gives to the terms 
which it separates. The proposition men are animals 
means that all the individuals, man, are contained among 



28 ON THE ALGEBRA OF LOGIC. 

the individuals, animal ; but that the qualities which dis 
tinguish an animal are contaiued among the qualities 
which distinguish a man. The proposition no stones are 
plants means that the objects denoted and the qualities 
connoted by the term stone are inconsistent with the 
objects denoted and the qualities connoted by the term 
plant. It is to be remembered that every term is at 
once a sum of objects and a product of qualities. If 
the term a denotes the objects a l9 2 > a z . . . and con 
notes the qualities a l9 2 , 3 . . . , then 



a = a l a 2 + . . . 
and the full content of the proposition no a is b is 



But the full content of the proposition all a is b can be 
expressed only by the two statements 

a 1 + a 2 +... + a i -< b L + b 2 + b s + .. . and ftft. . .&-< a^ag... 

where the i objects a are identical each with some one of 
the objects b and they qualities ft are identical each with 
some one of the qualities a. 

If p denotes a premise and c a conclusion drawn from 
it, then 

p V G (m) 

states that the premise and the denial of the conclusion 
cannot go together ; and 

p V o (n) 

states that the premise is sometimes accompanied by the 
falsity of the conclusion. It is hardly necessary to men 
tion that (w) is satisfied by either the truth of the con 
clusion or the falsity of the premise, and that (n) 
implies that both the premise and the negative of the 
conclusion must, at some time, be true. 



CXN" THE ALGEBRA OF LOGIC. 



29 



The word inference (or consequence) implies proceed 
ing in a definite direction in an argument, either from 
the premise to the conclusion, or from the negative of the 
conclusion to the negative of the premise. The argu 
ment p v c may be called an inconsistency. It is a 
form of argument into which the idea of succession does 
not enter ; it simply denies the possible co-existence of 
two propositions. An inconsistency between two propo 
sitions is equivalent to each of two equivalent conse 
quences, and a consistency to each of two equivalent 
inconsequences ; or 



c=p-<c = 



The copulas V and V with the symbol oo give means for 
expressing the total non-existence and the partial exist 
ence of expressions of any degree of complexity. Pro 
positions with the symbol do not occur in this algebra. 



(16 ) x v = "x does 
not, under any circumstances, 
exist." 

A universal proposition does 
not imply the existence of its 
subject; therefore x \7 = "x 
(if there is any x) is not 
non-existent," a proposition 
which is true whatever x may 
be. 



(16) x V oo = "x is at 
least sometimes existent." 

A particular proposition 
does imply the existence of its 
subject ; therefore x V = " x 
exists, and at the same time 
does not exist," a propo 
sition which is false whatever 
x may be. 



Since the symbol will not appear at all in proposi 
tions expressed with these copulas, it will not be neces 
sary to write the symbol co . I shall therefore express 
" there is no x " simply by x V- 



30 



ON THE ALGEBRA OF LOGIC. 



(170 a\Jb = aby 

abc \/=a\/bc = ca\/b = ... 

To say that no a is b is the 
same thing as to say that the 
combination ab does not ex 
ist. 



(17) aVb 
abc V = aV bc 

To say that some a is b is 
the same thing as to say that 
the combination ab does at 
least sometimes exist. 



The factors of a combination which is excluded or not 
excluded may be written in any order, and the copula 
may be inserted at any point, or it may be written at 
either end. The proposition abc y de may be read " abc 
is-not de," " cd is-not abe" " abe is-not do, that is, is 
either not d or not c," etc. Any 0, 1, 2, 3, 4, or 5 of the 
letters may be made the subject and the others the predi 
cate, and the positive or the negative universal copula 
may be used ; or there are in all 2.32, = 64, different 
ways of putting the above proposition into words. 

If a is a proposition, a \j states that the proposition is 
not true in the universe of discourse. For several pro 
positions, abc v means that they are not all at the same 
time true ; and the way in which they are stated to be 
not all at the same time true depends on the character 
of the universe. If it be the universe of the logically 
possible, then p l p 2 c \/ states that pi and p% may be 
taken as the premises and c the conclusion of a valid 
syllogism. It is the single expression in this system for 
a proposition which in the system of inclusions appears 
in the several forms 



from the premises the conclusion follows ; if the conclu 
sion is false, one at least of the premises is false ; from 
one premise may be inferred either the conclusion or the 
contradictory of the other premise, and from one pre 
mise and the contradictory of the conclusion follows the 



ON THE ALGEBRA OF LOGIC. 31 

contradictory of the other premise. If the universe 
which is understood is the universe of what is possible 
in accordance with the laws of nature, then ab V denotes 
that the simultaneous truth of a and b is a contradiction 
of those laws. That x and y stand in the relation of 
cause and effect may be expressed by xy \j. If a? is a 
certain position and y its attendant acceleration, the 
above proposition states that the position and the ab 
sence of the acceleration are not found together; that 
from the position may be inferred the acceleration, and 
from the absence of the acceleration may be inferred the 
absence of the position. If a V I means that Greeks are 
brave, and c \j d means that the megatherium is not 
extinct, then 

(a\/l)\/(c\j d) 

affirms that the co-existence of these two propositions is 
excluded from the universe of what is actually true. In 
like manner, according to the character of the universe 
of discourse, a V 6 denotes either that the two proposi 
tions are logically consistent, or that they are possibly 
co-existent, or that they have actually been at some mo 
ment of time both true. 1 

ALGEBRA OF THE COPULA. 
By the definition (1), we have 

(18) (a = b) = (ayt)(ayb). 
Since also 

(a = 5) = (*v!)(av), 

it follows that 

(19) ( a = b) = (a = I). 
In particular, 

(20) (ab = 0) = (ab = oo) = (ab y oo) ; 

1 The thorough-going extension of the idea of a limited universe to the 
relations between propositions is due to Mr. Peirce. 



32 ON THE ALGEBRA OF LOGIC. 

for the exclusions to which each equation is equivalent 
are (ab V oo) (db V 0), 

and ab V is a proposition of no content. 

The principles of contradiction and excluded middle 
are therefore completely expressed by 

(70 aa v- 1 (7) a+a V. 

In any symbolic logic there are three subjects for con 
sideration, the uniting and separating of propositions; 
the insertion or omission of terms, or immediate infer 
ence ; and elimination with the least possible loss of 
content, or syllogism. 

On uniting and separating Propositions. From the 
definitions of logical sum and logical product applied 
to terms and to propositions we have the following iden 
tities : 



(210 (a 

for the first member of the 
equation states that a does 
not exist and that b does not 
exist; and the second mem 
ber states that neither a nor 



(21) 

for the first member of the 
equation states that either a 
exists or b exists; and the 
second member states that 
either a or b exists. 



b exists. 

In both cases, a and b may be logical expressions of 
any degree of complexity. A combination of any num 
ber of universal propositions, or an alternation of any 
number of particular propositions, is then expressed as a 
single proposition by taking the sum of the elements of 
the separate propositions. This is the only form of in 
ference (if it should be called inference at all) in which 
the conclusion is identical with the premises. The equa 
tions (21 ) and (21) are not in reality two distinct 



ON THE ALGEBKA OF LOGIC. 



33 



equations ; they are, by (19), one and the same equa 
tion; since, by (13), the negative of (v) (^ V) i s 
(a V) + (ft V), and the negative of a + by is a+ ft V. 
They are each equivalent to the two inconsistencies, 

(a v) (b v) V + ft V) 
C J 

( v) + v) v + ^ v)- 

There is no single expression in this algebra for a sum 
of universal propositions or a product of particular pro 
positions. 



To express that the propo 
sitions, some a is b and some c 
is d, are not both at the same 
time true (or that it is true 
throughout the universe of dis 
course that either no a is b or 



To express that neither of 
the propositions, no a is b and 
no c is d, is true (or that it is 
true throughout the universe 
of discourse that both some a 
is b and some c is d), we must 
write 



else no c is d), we must write 

(a y ft) (c V d) y> 
And the expression for the corresponding particular 
propositions which follow from these universals is 

(a \7 ft) + (c v d) v ; 
that is, there is some time 



when either no a is b or else 
no c is d. 



(a V b) (c yd) V ; 

that is, there is some time 
when both some a is b and 
some c is d. 



On inserting and dropping Terms. The following in 
consistencies are immediate consequences of the defini 
tions of the sum and the product : 



(23) (abc V ) v (^ V)- 

The first asserts that the total non-existence of several 
things is inconsistent with the existence of some of 
them ; the second asserts that the total non-existence 



34 



ON THE ALGEBRA OF LOGIC. 



of something, as ab, is inconsistent with the existence of 
some part of it, as ab which is c. They are not two dis 
tinct inconsistencies, however ; either may be derived as 
a consequence from the other. These inconsistencies, 
when put into the form of inferences, become 

(22) If a + b V, 

then a + b + c V ; 

(23) If abc V, 
then ab V. 

That is to say, given a par 
tial inclusion, factors may be 
dropped and parts of a -sum 
may be introduced, but not 
without loss of content. 



(22 ) If a + b + c V? 
then a + b y > 

(23 ) If ab V> 

then abc y . 

That is to say, given a uni 
versal exclusion, factors may 
be introduced and parts of a 
sum may be dropped, but not 
without loss of content. 



As a particular case of both of the inconsistencies 
(22) and (23) we have 

(a yb)(cyd)y(ac\fb + d). 1 I. 

If into the expression which is affirmed not to exist, 
ab + cd, we introduce the factor c + a ; and if from the 
product, acb + acd + ab + cd, we drop the parts of a sum, 
ab + cd, there remains ac (6 + eT), the existence of 
which is inconsistent with the non-existence of ab and 
cd. Since there is no difference between subject and 

predicate, 

(ayb)(c~yd)y(a + c\f bd) 

is an inconsistency of quite the same nature as I. For 
the expression of /. in words we have 

I a . It is not possible that what is common to several 
classes should have any quality which is excluded from 

1 In its affirmative form, "if a is b and c is d, then ac is bd," this is 
Theorem I. of Mr. Peirce s paper on the Algebra of Logic (XXL). As 
pointed out by Mr. Venn, it was first given by Leibnitz : " Specimen de- 
monstrandi," Erdmann, p. 99. 



ON THE ALGEBRA OF LOGIC. 35 

one of them. If, for example, no bankers are poor and 
no lawyers are honest, it is impossible that lawyers who 
are bankers should be either poor or honest. 

In this way the theorem is put into words in terms of 
a quality which is excluded from a class. It is a pro 
perty of the negative copula that it lends itself equally 
well to the expression of propositions wholly in exten 
sion and wholly in intension, and also with the subject 
taken intensively and the predicate extensively. We 
should have in words, in these cases respectively 

I b . If several classes are respectively excluded from 
several others, no part of what is common to them can 
be included in any of the others ; 

I c . If several qualities are inconsistent respectively 
with several others, their combination is not consistent 
with any of the others. 

I d . It is not possible that a combination of several 
qualities should be found in any classes from each of 
which some one of those qualities is absent. If, for 
example, culture is never found in business men nor 
respectability among artists, then it is impossible that 
cultured respectability should be found among either 
business men or artists. 

The inconsistency I. is the most general form of that 
mode of reasoning in which a conclusion is drawn from 
two premises, by throwing away part of the information 
which they convey and uniting in one proposition that 
part which it is desired to retain. It will be shown that 
it includes syllogism as a particular case. The essential 
character of the syllogism is that it effects the elimina 
tion of a middle term, and in this argument there is no 
middle term to be eliminated. 

When combinations of any number of terms are given 
as excluded, a proposition with which they are inconsis- 



36 ON THE ALGEBRA OF LOGIC. 

tent can be formed by taking any number of terms out 
of each and uniting them as a sum and denying their 
co-existence with the product of the terms which re 
main. If _ _ 
abc v? plh V? 

affirm that no American bankers are uncharitable and 
that no Philadelphia lawyers are dishonest, then it is im 
possible that any Philadelphia bankers are either un 
charitable Americans or dishonest lawyers; that any 
uncharitable and dishonest lawyers are either Philadel- 
phians or American bankers ; that any bankers who are 
also Philadelphia lawyers are either uncharitable Ameri 
cans or dishonest, etc. Any, none, one, two, or three, 
terms from the first premise may be taken to form the 
sum with any, none, one, two, or three, terms from the 
second premise; there are, therefore, sixteen different 
conclusions to be drawn in this way from these two 
premises, of which dbcpTil \/ is the least, since it has 
dropped the most information, and abc + plh y is the 
greatest, since it has dropped none of the information. 

The inconsistency I. may be put into an inference in 
four different ways, according as both universals, one 
universal, one universal and the particular, or the parti 
cular alone, is taken as premise and the negative of what 
remains as conclusion. There are, therefore (when I. 
contains the smallest possible number of propositions), 
four distinct forms of inference, or progressive argu 
ment, with no middle term, in each of which the con 
clusion is a diminished conclusion. The factors of I. 
are, in general, one particular and any number of uni 
versals. The number of distinct forms of progressive 
argument which can be made out of an inconsistency 
between n propositions of which n 1 are universal, by 
taking 1, 2, . . or n 1 of the universal propositions with 



ON THE ALGEBRA OF LOGIC. 



37 



or without the particular proposition as premise and the 
negative of what remains as conclusion, is 2(^ 1). 
Argument by way of inconsistencies, therefore, what 
ever may be thought of its naturalness, is at least 
2 (n 1) times more condensed than argument in the 
usual form. 

When I. is made into an inference in such a way that 
one conclusion is drawn from two premises, we have, 



if the premises are both uni 
versal, 

(24 ) V * 

c \7 d 
. . ac \/ b rh d 

If no bankers have souls 
and no poets have bodies, then 
no banker-poets have either 
souls or bodies. 



if the premises are one uni 
versal and one particular, 

(24) ayb 

ac v # + d 
. . c v d 

If no Africans are brave 
and some African chiefs are 
either brave or deceitful, then 
some chiefs are deceitful. 



On Elimination. In (24 ) there is no elimination, 
and in (24) there is elimination of the whole of the 
first premise and part of the second. The most common 
object in reasoning is to eliminate a single term at a time, 
namely, one which occurs in both premises. Each of 
these inferences gives rise to a form of argument, as a 
special case, by which that object is accomplished, 
the premises being on the one hand both universal, and on 
the other hand one universal and the other particular. 
The inconsistency I. becomes, when d is equal to 5, and 
hence b + d equal to oo, 

(a v V) (c V &) (ac v <*>) V> 
or 

(a v 1) (^ v c ) ( G V a ) V- -^ 

Given any two of these propositions, the third proposi 
tion, with which it is inconsistent, is free from the term 



38 



ON THE ALGEBRA OF LOGIC. 



common to the two given propositions ; a, &, and c are, 
of course, expressions of any degree of complexity. 
The propositions ma ^/x + y,xy~\jc + n, for instance, 
arc inconsistent with ma V c + n; any number of terms 
may be eliminated at once by combining them in such a 
way that they shall make up a complete universe. 

When any two of the inconsistent propositions in II. 
are taken as premises, the negative of the remaining one 
is the conclusion. There are, therefore, two distinct 
forms of inference with elimination of a middle term, 
special cases of (24 ) and (24). If we write x for the 
middle term, we have 



(25 ) a\/ x 

b\/x 
.-. ab v- 

The premises are 
a (b + 1) x \/ 
(a + a) bx v ; 
and together they affirm that 



or 



ab (x + x) + abx + abx v> 

ab + abx + ax V- 

Dropping the information con 
cerning x, there remains 

ab y. 

The information given by the 
conclusion is thus exactly one 



half of the information given 
by the premises (Jevons). 



(25) a v x 

bV x 
.-. ba V. 
The second premise is 

bx (ax+ax) V, 
which becomes, since there is 



no ax, 



or 



bx (a + x) V, 
bxa\f. 



Dropping the information con 
cerning x, there remains 

baV. 

This conclusion is equivalent 
to 



ba V x + #; 

but the premises permit the 
conclusion 

ba V z; 

hence the amount of informa 
tion retained is exactly one half of the (particular) infor 
mation given by the premises. 



ON THE ALGEBRA OF LOGIC. 



39 



Elimination is therefore merely a particular case of 
dropping irrelevant information. 

When a and b are single 
terms, (25 ) is the doubly uni 
versal syllogism, and it is the 
single form in which that 
syllogism appears in this alge 
bra. When it is translated 
into syllogism with an affirm 
ative copula, it is necessary 
to consider the four variations 
of figure which are produced 
according as x or x is made 
subject or predicate. The 
eight moods in each figure 
correspond to the eight varia 
tions of sign which may be 
given to a, b. and x. All the 
rules for the validity of the 
doubly universal syllogism are 
contained in these : 



(1) The middle term must 
have unlike signs in the two 
premises. 

(2) The other terms have 
the same sign in the conclu 
sion as in the premises. 



When a and b are single 
terms, (25) is the universal- 
particular syllogism, and it 
is the only form of that syllo 
gism in this algebra. It can 
be translated into eight differ 
ent forms of syllogism with 
unsymmetrical copula, accord 
ing as x is made subject or 
predicate of either premise, 
and according as a or b is 
made subject of the conclu 
sion. The eight moods of the 
major and minor particular 
syllogism in each figure corre 
spond to the eight variations 
of sign which may be given 
to a, b, and x. All the rules 
for the validity of the uni 
versal-particular syllogism are 
contained in these : 

(1) The middle term must 
have the same sign in both 
premises. 

(2) The other term of the 
universal premise only has its 
sign changed in the conclu 
sion. 



Those syllogisms in which a particular conclusion is 
drawn from two universal premises become illogical 
when the universal proposition is taken as not implying 
the existence of its terms. 1 



1 McColl : Symbolical Reasoning, Mind, no. xvii. Peirce : Algebra 
of Logic, Am. Journal of Math., vol. iii. 



40 ON THE ALGEBRA OF LOGIC. 

The argument of inconsistency, 

(a v 1) (J> V <0 V ) v> II- 

is therefore the single form to which all the ninety-six 
valid syllogisms (both universal and particular) may be 
reduced. It is an affirmation of inconsistency between 
three propositions in three terms, such that one of the 
propositions is particular, and the other two are univer 
sal ; and such that the term common to the two universal 
propositions appears with unlike signs, and the other two 
terms appear with like signs. Any given syllogism is 
immediately reduced to this form by taking the contra 
dictory of the conclusion, and by seeing that universal 
propositions are expressed with a negative copula and 
particular propositions with an affirmative copula. Thus 
the syllogisms Baroko and Bokardo, 1 

All P is M, Some M is not P, 

Some S is not M, All M is , 

.-. Some S is not P, /. Some S is not P, 

are equivalent respectively to the inconsistencies 

(P V M) (S V M) (S V P) V, 
(MM P) (MV S) (S V P) V. 

1 If there were ever any occasion to use the mnemonic verses of syllo 
gism, it might be worth while to put them into a form in which each word 
should bear the mark of its figure, as well as of its mood and its method 
of reduction. By some slight changes in the words, the first, second, 
third, and fourth figures might be indicated by the letters r, t, I, and n 
respectively : 

(r) Barbara, Cegare, Darn, Ferioque prioris. 

(t) Cesate, Camestes, Festive, Batoko secundee. 

(I) Tertia, Dalipi, Disalmis, Dalisi, Felapo. 

(I) Bokalo, Feliso, habet ; quarta insuper addit, 

(n) Bamanip, Camenes, Dimanis, Fesanpo, Fesison. 



ON THE ALGEBRA OF LOGIC. 41 

It is then possible to give a perfectly general rule, easy 
to remember and easy of application, for testing the 
validity of any syllogism, universal or particular, which 
is given in words. It is this : 

Rule of Syllogism. Take the contradictory of the con 
clusion, and see that universal propositions are expressed 
with a negative copula and particular propositions with 
an affirmative copula. If two of the propositions are 
universal and the other particular, and if that term only 
which is common to the two universal propositions has 
unlike signs, then, and only then, the syllogism is valid. 

For instance, the syllogism 

Only Greeks are brave, 
All Spartans are Greeks, 
Therefore all Spartans are brave, 

is equivalent to the inconsistency 

Non-Greeks are-not brave, 
Spartans are-not non-Greeks, 
Some Spartans are not-brave, 

which fails to stand the test of validity in two respects, 
the term brave appears with unlike signs and the term 
Greeks with like signs. The syllogism - 

All men are mortal, 
Some mortals are happy, 
Therefore some men are happy, 

is equivalent to the inconsistency 

Men are-not immortal, 
Some mortals are happy, 
Men are-not happy, 

and it is not valid for the same reasons as before, the 



42 ON THE ALGEBRA OF LOGIC. 

term mortal appears with unlike signs, and the term 
men with like signs. 

When #, 5, and x are expressions of any degree of 
complexity, (25 ) and (25) still furnish the only means 
for the elimination of x. For instance, if 

(ab -f cd) x v 

and 

(a + c)x + bfy, 
then 



or 

abc + dcd + bf y, 

is all that can be said without reference to x. And if 

(ab + cd)x + bfy 
and 

(a + c)x y } 

then the conclusion, irrespective of #, is 



(ab + cd) a + c + bf y, 
or 

ac b 



If the premises consist of propositions about proposi 
tions, then any proposition which it is desired to drop 
may be eliminated in accordance with these two rules. 

Syllogisms are the inferences, with elimination, which 
are obtained by taking two of the propositions of I. as 
premises and the other as conclusion. When one propo 
sition only is taken as premise, the conclusion is an 
alternation of propositions ; and, as a special case, a 
single arbitrary term (instead of two or none) may be 
introduced. We have 



ON THE ALGEBKA OF LOGIC. 



43 



ayb, 



(26 ) 



or, in words, if no a is b, then 
either no ac is either b or d, 
or else some c is d. If no 
Africans are brave, then either 
some chiefs are deceitful, or 
else no African chiefs are 
either brave or deceitful. 
When c = x, d l, this be 
comes 
(27 ) _v* 

.. (a v x) + (5 V x )- 
If no Africans are brave, then 
either no Africans are Chinese 
or else some Chinese are not 
brave. 



(26) 



acVb + d , 



or, in words, if some ac is 
either b or d, then either some 
a is b or some c is d. If some 
African chiefs are either brave 
or deceitful, then either some 
Africans are brave or some 
chiefs are deceitful. When 
b = d = x, this becomes 

(27) ac V 

.-. (a v a;) + (c V ac). 

If some lawyers are bankers, 
then either some lawyers are 
honest or some bankers are 
dishonest. 



Inference from Universal to Particulars. Dimin 
ished statement and that particular form of diminished 
statement which is syllogism are the only reasoning pro 
cesses that are valid when a universe which contains 
nothing is included among possible universes, that is, 
when it is taken as possible that both x and x may be at 
the same time non-existent. When that universe is ex 
cluded, when the postulate "z and non-z cannot both 
be non-existent" is taken as true, one other form of 
reasoning is possible. That postulate is expressed by 



(x v) V ( x V)> 



P. 



which is equivalent to the two inferences, " if x does not 
exist, then non-z does exist," and " if non-z does not 
exist, then x does exist ; " or, from the total non-exist 
ence of any expression whatever may be inferred the 
existence of some part at least of its negative. If 



44 ON THE ALGEBRA OF LOGIC. 

a(b+c)v, then + 5c V , and iid + le y , then a (6 + <?) V ; 
or, 



If # is a proposition, & V 5, then non-# is its denial, 
a\/b , and the postulate states that a proposition cannot 
be both true and false at the same time. 
From the proposition 

ab v 
follows, in this way, 

ab v > that is, a + 5 V. 

The complete convention in regard to the existence of 
terms is therefore : the particular proposition a V b im 
plies the existence of both a and b ; the universal propo 
sition a y b does not imply the existence of either a or 5, 
but it does imply the existence of either a or b. The 
necessity of the convention (if it should be called a con 
vention) is even more evident when a and b are proposi 
tions ; in that case it is equivalent to saying that two 
propositions cannot be true together unless each is at 
some time true, and that they cannot be not true to 
gether unless one or the other is at some time false. 

Mr. McColl has pointed out that from u all a is 5," 
" some a is b " does not follow, because there may not 
be any a. But from 

aB v 
it does follow that 

a5V ; that is, ab + ab + ab V ; 

or from " all a is b " it does follow that one at least of 
the propositions " some a is 6," " some not-a is 5," 
" some not-a is not 5," is true. From any universal prop 
osition follows some one at least of the three particular 
propositions which it does not contradict. If a is known 



ON THE ALGEBRA OF LOGIC. 45 

to exist, then " some a is b" follows from " all a is b " 

by a syllogism : 

aB v 

aaV 
. . ab V 

From " no sea-serpents have gills " we cannot infer that 
there are some sea-serpents which are without gills, un 
less it is known that there are some sea-serpents ; but 
we can infer that either there are some sea-serpents 
without gills, or there are some things, with or without 
gills, which are not sea-serpents, or else there is nothing 
in the universe. 

EESOLUTION OF PROBLEMS. 

Rule. Express universal propositions with the nega 
tive copula and particular propositions with the affirma 
tive copula, remembering that a b is equivalent to 

ab + ab Y, 

and that its contradictory, a is not equal to 6, is equiva 
lent to 

al + ab V. 

From a combination of universal propositions, the con 
clusion, irrespective of any term or set of terms to be 
eliminated, x, consists of the universal exclusion of the 
product of the coefficient of x by that of the negative 
of x, added to the excluded combinations which are free 
from x as given. If the premises include an alternation 
of particular propositions, the conclusion consists of the 
partial inclusion of the total coefficient of x in the par 
ticular propositions by the negative of that of x in the 
universal propositions, added to the included combina 
tions which are free from x as given. 



46 ON THE ALGEBRA OF LOGIC. 

If there is any reason for expressing a universal 
conclusion with an affirmative copula or a particular 
conclusion with a negative copula, it can be done by 
taking any term or set of terms as subject and the 
negative of what remains as predicate. 

The premises may also contain an alternation of any 
number of universal propositions. If either 

(p v x) or (q v x) or (r y z), 
and if at the same time 

am y x, 
then 

am (p + q + rz) V 

is the conclusion irrespective of x. When a combina 
tion of particular propositions is included among the 
premises, the conclusion consists of a combination of 
the same number of particular propositions. From 

(pyx) fev^) 
(a Vx) (bV x), 

may be inferred the two propositions, 

(ay fq) (by pq). 

From particular propositions by themselves no con 
clusion follows, otherwise than by simply dropping un 
necessary information. 

Particular premises may be attached to the universal 
premises by the conjunction or instead of the conjunction 
and. In that case no elimination is possible (except 
what can be done between the universal propositions by 
themselves), and a conclusion can be obtained only by 
means of "the postulate, P. If either (ayb and c \j cT) 
or (jg v h and i vy), then the conclusions are gh + ab y, 
ij + ab V, gli + cd V, ij + cd V. In general, then, the 
premises may consist of a combination or an alter- 



ON THE ALGEBRA OF LOGIC. 47 

nation of universal propositions (two cases), or of par 
ticular propositions (two cases), or a combination or 
an alternation of universal propositions united as a sum 
or a product to a combination or an alternation of 
particular propositions (eight cases). 

It is apparent that logical notation would be improved 
by the addition of another sign, by means of which an 
alternation of universal and a combination of particular 
propositions might be expressed as a single propo 
sition, a sign such that 

(p + x) sign qy sign rz \/ 

should mean that some one of the expressions p + x, qy, 
rz, is totally non-existent, and its contradictory, 

(p + x) sign qy sign rz V, 

should mean that all of these are, at least in part, 
existent. 

The plan of treating a set of universal premises as a 
command to exclude certain combinations of the terms 
which enter them is due to Boole ; no adequate exten 
sion of his method so as to take in particular propo 
sitions is possible, without the use of some device which 
shall be equivalent to a particular copula. Boole s 
method of elimination between universal propositions 
is to put x first equal to and then to 1 in the given 
function, and to take the product of the results so ob 
tained. The only difference between this rule and that 
which I have given (which is Prof. Schroder s) is that 
it first introduces x into those terms which are already 
free from it, and then proceeds to eliminate it from all. 
The value of the function 

ax _|_ ix + c, or ax + bx + c (x + x), 
for x (in this case b + c) is the coefficient of x, and 



48 ON THE ALGEBRA OF LOGIC. 

its value for x 1 (in this case a + c) is the coefficient 
of x. I have shown that the method is not an invention 
of modern times, but that it is nothing more than a rule 
for working the syllogism, 

All b is x, No a is x, .-. No a is.ft, 

when a, 5, and x are not restricted to being simple 
terms. With the unsymmetrical copula, there are four 
different forms of pairs of universal propositions which 
make possible the elimination of x (XXI., p. 39), and 
for its elimination between a universal and a particular 
proposition it would be necessary to consider eight 
different forms, corresponding in all to the twelve dis 
tinct forms of syllogism. 

If the result which remains after elimination is of the 
form 

am + bm + c y (c) 

(where m is the term in regard to which information is 
sought, and where all the letters are expressions of 
any degree of complexity), and if there is any reason for 
being dissatisfied with the conclusion as it stands, 
" no m is a, no b is not m, and there is no c," m may 
be made subject and predicate respectively of two affirm 
ative propositions, " all b is w, and all m is a." If it 
be desired to express the conclusion without any repe 
tition, then we must first state what is true without 
regard to m, in this case, 

ab + c \7> 

"there is no ab nor <?," and then this information 
must be used to diminish the propositions in m. The 

identities 

a = a (ab + c + ab + c) 



b = b (ab + c + ab + c) 
become, when there is no ab + <?, 



(8 ) 



ON THE ALGEBKA OF LOGIC. 49 

a = a . ab + c = #5c, 
b = b . ab + c = bac ; 
and hence, instead of 

a v m>t by m, 
it is sufficient to write 

ale v m>) boo y m ; 
or, affirmatively, 

All m is b + c + a, 
All bac is m. 

Prof. Schroder expresses in terms of m such a con 

clusion as 

am + &?H + c (m + m) = 0, 

by means of the formula 

[0 + c) m + (b + c) m = 0] 
= [m = all (b + c) + some a + c] [&& + c = 0]. 

The first factor of the second member of the equation is 
equivalent to the propositions, 

All m is b + c + ac, 
All (6 + c) is w, 



Some a + c is m ; 

that is, it contains the propositions of the first member 
(the first diminished by ab + c = and the second not), 
but it contains in addition the particular proposition 
" some a + c is wi," which is a legitimate inference 
from " no (a + c) is m" only if a + c is known to exist. 
A more condensed equational form of the conclusion 
am + bm + c\/ is 

(m = all bac + some TJac) (ab + c = 0). 
Boole reaches the same conclusion, ((7), but he does 



50 ON THE ALGEBKA OF LOGIC. 

it by an extremely circuitous route. Nothing could well 
be simpler of application or more evident than this 
rule of Prof. Schroder s, and there is no reason why 
one should not place implicit confidence in it, in an 
algebra in which particular propositions are not taken 
as implying the existence of their terms. It contains 
the solution of what Mr. Jevons calls the " inverse log 
ical problem," and which he solves by a process " which 
is always tentative, and consists in inventing laws and 
trying whether their results agree with those before us " 
(XXII., p. 252). It makes all reference to tables and 
machines quite superfluous. It seems to have been 
overlooked by the latest expositor of Boole s system, 
Mr. Venn. He says that Boole s method of getting his 
conclusion is " a terribly long process ; a sort of ma 
chine meant to be looked at and explained, rather than 
to be put in use ; " and that if ever we do feel occasion 
to solve such a problem, it can be done most readily 
" by exercise, so to say, of our own observation and 
sagacity, instead of taking, and trusting to, a precise 
rule for the purpose of effecting it " (XXIII., p. 316). 

But Boole s form for the conclusion (besides being 
not quite legitimate in this algebra) is not that which 
is most natural or most frequently useful. It is, more 
over, suited only to a logic of extension, and it would 
be difficult to interpret intensively. The very simple 
device which may be substituted for ,it is to make use 
of the same method for getting back from excluded 
combinations to affirmative propositions which was em 
ployed in passing from the given affirmative propositions 
to the excluded combinations : if 

All b is m = b\/ m, 
then 

b m = all 1) is m. 



ON THE ALGEBRA OF LOGIC. 51 

In this way the conclusions are given in the form which 
has been adopted by Mr. McColl. Complicated prob 
lems are solved with far more ease by Mr. McColl than 
by Mr. Jevons ; but that is not because the method of 
excluded combinations is not, when properly treated, 
the easiest method. A method of implications, such 
as that of Mr. McColl, is without doubt more natural 
than the other when universal premises are given in 
the affirmative form, but the distinction which it pre 
serves between subject and predicate introduces a rather 
greater degree of complexity into the rules for working 
it. An advantage of writing abc y instead of dbc = 
is that the copula can be inserted at any point in the 
excluded combination, and that elimination can be per 
formed on the premises as they are given, when they 
have been expressed negatively, without first trans 
posing all the members to one side. Without some 
thing corresponding to a contradictory copula, particular 
propositions cannot be treated adequately, and compli 
cated propositions of either kind cannot be simply 
denied. With it, the contradictory of " all a is all ," 
that is, " it is not true that all a is all &," is al + db V ; 
that is, " either some a is not b or some b is not a." 
And the contradictory of 

abc + abc + abc y 
is 

abc + abc + abc V ; 

that is, some one at least of the given combinations is 
in existence. 

EXAMPLES. 

1. (By Mr. Venn in Mind for October, 1876.) The 
members of a board were all of them either bondhold 
ers or share-holders, but no member was bond-holder 



52 ON THE ALGEBRA OF LOGIC. 

and share-holder at once ; and the bond-holders, as it 
happened, were all on the board. What is the relation 
between bond-holders and share-holders ? 

Put 

a = member of board, 
b bond-holder, 
c = share-holder. 

The premises are evidently 

a v be + %Cj 
bya; 

and taking the product of the coefficient of a by that of 
a, we have 

b (be + 5c) Y> 
or 

bey. 

The required relation is, therefore, 

No bond-holders are share-holders. 

2. (XXII., p. 283.) What are the precise points of 
agreement and difference between two disputants, one 
of whom asserts that (1) space (a) = three-way spread 
(5), with points as elements (<?) (Henrici) ; while his 
opponent holds that (2) space = three-way spread, and 
at the same time (3) space has points as elements ? 

(a = be) = (aB + ac + abc \/), (1) 



ac y. (3) 

They both assert that 

a5 + ac -f- #c v? 

and the second asserts in addition that 

dbc Y j 



ON THE ALGEBRA OF LOGIC. 53 

that is, that a three-way spread which had not points as 
elements would be space. 

3. (XVI., vol. x. p. 21.) From the premises 



bxyc (cl + i/)e 
ab v x (3, + e) c 

a + b + 



deduce a proposition containing neither x nor y. 

The term y does not occur at all ; hence y can be 
eliminated only by dropping the parts which contain it. 
There remain 

acct + alj (<l + e) y x, 

bcde v x ; 
and taking the product of the first members we have 

abcde \/. 

4. (XXIIL, p. 310.) 

Given ^ ~ > , find xz in terms of a and c. 
yz = c ) 

The equations are equivalent to the exclusions 

xya -\-xa-\-ya v? 
yzc -\-yc + zc\/; 
and after elimination of y there remains 

xa + zc + %ac + z<w V- O 9 ) 

Collecting the predicates of xz and xz, we have 

xz v dc + c, 

(?) 
ic + * V ac 



54 ON THE ALGEBEA OF LOGIC. 

Prof. Schroder s formula, (7, p. 49, 

If m v x and m y y> then m = all y + some x, 
gives, in this case, 

xz all ac + some (ac + ac) 

= all ac + some do. 
If it were required to find xz + xz, we should have 

xz -\-xz\/ ac, 
xz + xz v ac + ac ; 
whence 

cci + ieg = all (ac + ac) + some (ac + ca + ac) 

= all (ac + ac) + some ac. 
It is evident that (jp) cannot be inferred from (g). 

5. (Educational Times, Feb. 1, 1881, 6616. By W. B. 
Grove, B. A.) The members of a scientific society are 
divided into three sections, which are denoted by a. b, c. 
Every member must join one, at least, of these sections, 
subject to the following conditions : (1) Any one who 
is a member of a but not of 5, of b but not of c, or of c 
but not of a, may deliver a lecture to the members 
if he has paid his subscription, but otherwise not; 
(2) one who is a member of a but not of c, of c but not 
of a, or of b but not of a, may exhibit an experiment 
to the members if he has paid his subscription, but 
otherwise not; but (3) every member must either 
deliver a lecture or perform an experiment annually 
before the other members. Find the least addition to 
these rules which will compel every member to pay his 
subscription or forfeit his membership, and explain the 
result. 



ON THE ALGEBRA OF LOGIC. 55 

Put x = he must deliver a lecture, y = he must per 
form an experiment, and z = he has paid his subscrip 
tion. Then the premises are 

ale v (a) 

al + Ic + cd v xz (1) 

ac + cd + db y yz (2) 

xyy. (3) 

It is required that z be excluded from all that part of 
the universe from which it has not already been ex 
cluded ; namely, from the negative of 

(al + be + cd) x + (ac + ca + ab) y + ale + xy, 
which is, by the second rule for getting the negative, 

(ale + abc + x) (ale + ac + y) (a + b + c) (x + ?/), 
or 

abcx + ac^y. 

Hence the desired " least addition to the rules " is 
abcx -f- acxy y z, 

or, " No one who has not paid his subscription can be 
a member of all three sections and deliver a lecture, 
or of a and c and perform an experiment without lec 
turing." 

6. (III., p. 237. Proposed for simpler solution by 
Mr. Grove, Educational Times, April 1, 1881.) A num 
ber of pieces of cloth striped with different colors were 
submitted to inspection, and the two following observa 
tions were made upon them : 

(a) Every piece striped with white (w~) and green (#) 
was also striped with black (5) and yellow (?/), and 
vice versa. 



56 ON THE ALGEBRA OF LOGIC. 

(5) Every piece striped with red (d) and orange (r) 
was also striped with blue (w) and yellow, and vice 
versa. 

It is required to eliminate yellow, and to express the 
conclusion in terms of green. 

The premises are 

W 9 fy> dr = uy ; 

and by (18 ) they are equivalent to the exclusions 



dr (u + y) + uydr y. 

Collecting the coefficients of y and y we have 
bwg + udr y y, 
Wff + dryy; 

and taking the product of the left-hand members we 
have 

uwgdr + bdr (w + g) y, 

which is to be added to that part of the premises which 
does not contain y ; that is, to 

wig + dru y. 
Concerning g we have 

g y w (5 + udr), bdr y g ; 

or, with the affirmative copula, by (30), 

g -< w + bu + bdr, bdr -< g. 

The first is equivalent to Boole s conclusion when that 
is reduced by dru = 0. For the second Boole gives only 
bdrwu -< g. 

To solve this problem by Mr. Jevons s method, it 
would be necessary to write out the one hundred and 
twenty-eight possible combinations of seven terms, and 
to examine them all in connection with each of the 



ON THE ALGEBRA OF LOGIC. 57 

premises. As Mr. Jevons himself says : " It is hardly 
possible to apply this process to problems of more than 
six terms, owing to the large number of combinations 
which would require examination" (XIII., p. 96). 

7. (III., p. 146). From the premises 

xz (v + wy + wy) \f 

v xw (yz + yz) \J 
x(v + y) (zw + zw) V 
(x + vy) (zw + zw) V 

it is required, first, to eliminate v ; second, to express 
the conclusion in terms of x ; third, in terms of y ; 
fourth, to eliminate x m , fifth, to eliminate y. 
The terms which involve v are 



xz + xw (yz + yz) + y (zw + zw) y v, x (zw + zw) y v ; 

whence, taking the product of the left-hand members, 
we have only 

xzyw v, ( a ) 

which is to be added to that part of the premises which 
does not contain v, namely, to 

xz (wy + wy) + xy (zw + zw) + x (zw + zw) \/. 
Collecting the parts which contain x and x we have 

x \7 zw + yzw, (b) 

x \7 zw + zw + zwy. (c) 

The negative of the second member of (<?) is, by (14), 
zw + zwy^ hence, by (18 ), these two exclusions are 
equivalent to the identity 

x = zw + zw + zwy) (^) 

or 

x = zw + yzw. 



58 ON THE ALGEBRA OF LOGIC. 

No part of the conclusion lias been dropped in .(ft) 
and (c) ; hence the propositions which concern y may be 
taken from them. They are 



y xzw, xzw V $ 

or 

y -< x + z + w, xzw 

These exclusions yield nothing upon the elimination 
of y ; hence the only relation between a?, s, and w is, 
from (5) and (c), 

cezw; + Jezw + 5J# y". (y) 

These conclusions are the same as those of Mr. McColl, 
and they are equivalent to those of Boole and Schroder. 
Prof. Wundt (XVIII., p. 356) accidentally omits (a) in 
getting the conclusions in regard to y, and they are in 
consequence altogether wrong. He remarks that Schro 
der has treated the problem in a partly coincident 
manner. I do not find that Mr. Jevons has treated it 
at all. 

8. Six children, #, b, c, d, e, /, are required to obey 
the following rules: (1) on Monday and Tuesday no 
four can go out together ; (2) on Thursday, Friday, 
and Saturday, no three can stay in together; (3) on 
Tuesday, Wednesday, and Saturday, if b and c are to 
gether, then a, b, e, and / must remain together ; (4) 
on Monday and Saturday b cannot go out unless either 
d, or c, e, and / stay at home, b and/ are first to decide 
what they will do, and c makes his decision before , 
d, or e. Find () when c must go out, (/3) when he 
must stay in, and (7) when he may do as he pleases. 

Let a be the statement that a goes out, and a the 
statement that he stays in, etc. Then we have for the 
first two premises 



ON THE ALGEBRA OF LOGIC. 59 

M+ T\J abed + bcde + . . . (1) 

Th + F+ S v ale + aid + . . . (2) 

The third premise excludes from certain days the com 
bination in which b and c are both out or both in, ex 
cept when a, 6, e, and/ are together ; that is, 



T+W+ S~y(bc + bc) abef+ abej 
V (T>c + bc) (a + 5 + 



or, finally, 

T+W+ S\/tca + bce + bcf+ bca + bee + bcf. (3) 

The last premise is, for Monday, 

M~ybd(c + e+f). (4) 

On Saturday, c, e, and / cannot all stay at home, by 
(2) ; therefore, this part of the premise is 

Sybd. (4 ) 

The first thing required is the elimination of a, d, and 
e. That part of the premises which is already free from 
those letters is 



(3 ) 

Nothing can be eliminated between (1) and (2), because 
MTh = 0, etc. 

For the same reason, d cannot be eliminated between 
(4) and (2) ; and therefore the premise (4) must bo 
simply dropped, a and e can be eliminated at once by 
combining (3) with (1) and with (2). From (3) and 
(1), we have respectively 

(T + W+ S)bcya + e, 



60 ON THE ALGEBEA OF LOGIC. 

and taking the product of the right-hand members and 
the sum of the left-hand members, we have 

T(4o)v. (5) 

From (3) and (2) we have respectively 

(T+W+ S)Zc ya + e, 



whence, in the same way, 

S(lo) v. (6) 

By combining (4 ) with that part of (2) which does not 
contain a, e, or 25, and does contain cl, namely, with 



we obtain 

Sybfc. (7) 

The conclusion required is then contained in (2 ), (3 ), 
(5), (6), and (7). But the information given in regard 
to S and T may be somewhat simplified by collecting 
their predicates. We have 

S\/lcf+ lcf+ Ic + bcf+ be/, 
or 

Sytc + bf, (8) 

and 

Sr/5 (9) 



which with 

Th + fytcf, (2") 

Wybcf+lcf, (3") 

form the entire conclusion. Collecting the subjects of 
c and c } we have 

(Th + F) lf+ (T+ W) lf+ Stye (a) 

Tb + Wbfy c (b) 



ON THE ALGEBRA OF LOGIC. 61 

where the last proposition is already independent of c, 
and where c cannot be eliminated between (#) and (6). 
The conclusion may be expressed in words in this way : 
(a), if on Thursday or Friday b and / are both at 
home, or if on Tuesday or Wednesday / goes out with 
out 5, or if b stays at home on Saturday, then c must 
go out; (/3), if b goes out on Tuesday, or if b goes out 
without / on Wednesday, then c must stay at home ; 
(7), whether c goes out or stays in, b does not go out 
without /on Saturday. 

OX THE CONSTITUTION" OF THE UNIVERSE. 

The number of combinations in the complete develop 
ment of n terms is 2 n . In any actual universe of things, 
any one of these combinations may be either present or 
absent ; hence the number of different ways in which a 
universe may be made up out of n things is 2 2W . The 
following Table gives the sixteen possible constitutions 
of the universe with respect to two terms. The sign 1 
indicates the presence of the combination at the head 
of which it stands, its absence. With the aid of 
the dual notation, applied to logical algebra by Mr. 
Franklin, 1 each case may be defined by a number ; it is 
only necessary to attribute powers of two as weights to 
the different combinations, and to describe each arrange 
ment by the sum of the weights of the combinations 
which are present in it. If we take the, combinations of 
a and b in the order a&, db, al,dl, then 4, or 0100, de 
notes that the combination aB is present, and nothing 
else ; 9, or 1001, that dl and ab are present and al and 
db are absent, etc. 

1 Johns Hopkins University Circular, April, 1881. 



62 



ON THE ALGEBRA OF LOGIC. 



al 
8 


aH 
4 


ab 
2 


ab 

1 




























1 


1 








1 





2 








1 


1 


3 





1 








4 





1 





1 


5 





1 


1 





6 





1 


1 


1 


7 


1 











8 


1 








1 


9 


1 





1 





10 


1 





1 


1 


11 


1 


1 








12 


1 


1 





1 


13 


1 


1 


1 





14 


1 


1 


1 


1 


15 



If a is animal and b is black, then the 5th case is that 
of a universe made up of black animals and animals 
which are not black ; in the 12th case the things which 
are wanting are black animals and black things which 
are not animals, that is, there are no black things in 
this universe ; the 15th case is the actual universe with 
respect to the terms animal and black ; the 0-case is a 
universe in which nothing exists. If the material uni- 



ON THE ALGEBRA OF LOGIC. 63 

verse is the subject of discourse, and if a means matter 
and I means indestructible, then the existing state of 
things is described by 4 ; indestructible matter exists, 
and what is not indestructible matter docs not exist. 
This Table is given by Jevons (XIII., p. 135) ; but he 
does not take account of non-existent terms, and hence 
all but seven of the sixteen cases (all but 6, 7, 9, 11, 
13, 14, 15) are considered by him to be logical absurdi 
ties. If a and b are propositions, then case 9 is a 
universe in which they are true together and false to 
gether, and in which the time during which a is true 
is identical with the time during which b is true, either 
logically or extra-logically. The 0-case is a universe in 
which no proposition is true. Two cases the sum of 
whose characteristic numbers is 15, as 5 and 10, or 
0101 and 1010, have been called by Prof. Clifford 
complementary cases : what exists in one is what does 
not exist in the other. 

To exactly define the constitution of any universe, it 
is necessary to state, in regard to each combination, that 
it is present or that it is absent. The simple laws which 
every two terms obey are therefore four in number, 
being partly universal propositions and partly particu 
lar ; except in the 0-case, where all the universal propo 
sitions are true, and in case 15, where all the particular 
propositions are true. The perfectly symmetrical uni 
verses are thus the universe in which there is nothing 
and that in which there is some of everything. For 
case 8, we have 

(a \/b)(a\fb)(ayP)(ay ), 
and for case 13 

(a y b) (a v b) (a y 5) (u V I). 
When two simple or compound statements cannot be 



64 



THE ALGEBRA OF LOGIC. 



converted into each other by any interchange between 
the terms which enter them (including negatives of 
terms), they are said to belong to different types. The 
universal propositions in two terms are of six different 
types. None, one, two, three, or four of them may be 
true, and it is only in the case where two are true that a 
difference of type is produced by the way in which the 
propositions are selected. Those two may be taken so 
that one letter has the same sign in both or not. Thus 
we may have either, 

ab + ab y, 
that is, 



or 
that is, 



ab + ab \/, 



a . 



The following Table gives the six types, the proposi 
tions which define them, and the universes which belong 
to each type : 



Type. 


Universal. 


Particular. 


Cases. 


I. 




(a V ) (a V b) (a V 5) (a V 5) 


15 










II. 


a\/b 


(a V V) (a V I) (a V I) 


8, 4, 2, 1 


III. 


a v 


(a Vb)(aV I) 


12, 3, 10, 5 


IV. 


a = b 


(a Vb)(aV I) 


6, 9 


V. 


a-\-b\/ 


a\fl 


7,11,13,14 


VI. 


$,_{_ < 2_L--L-7J\7 




1 











I. and VI. are complementary types ; and so are II. and 
V. The universes complementary to III. and IV. are 



ON THE ALGEBRA OF LOGIC. 65 

of types III. and IV. respectively. Six is the number 
of types of a universe in two terms, when all the par 
ticular propositions which the universal propositions do 
not deny are known to be true. If one takes account of 
combinations of alternations and alternations of com 
binations of both particular and universal propositions, 
the number of types is largely increased. 

A race of beings which always completely defined its 
universe would have the above four-fold statements for 
its forms of expression. The eight propositions which 
are used by the race which exists are not complete 
definitions of a universe, but they are symmetrical; 
each has an eight-fold degree of ambiguity. "No a is 5" 
denies the existence of the combination a5, but it leaves 
it doubtful whether, of the remaining combinations, none, 
any one, any two, or all three exist. " Some a is 5," 
which affirms the existence of the combination ab, re 
stricts the universe to some one of the eight cases, 1, 
3, 5, 7, 9, 11, 13, 15. If, however, propositions are 
taken in the other sense, if positive (affirmative) 
propositions are taken as implying the existence of the 
subject and negative not, then they do not include all 
possible states of things with symmetry. The negative 
universal and the positive particular propositions cover 
eight cases each, as before ; but of the positive universal 
a v I takes in the four cases 1, 3, 9, 11, and a \j b the 
six cases 1, 4, 5, 9, 12, 13 only, and their contradictories, 
the negative particular, have respectively a twelve-fold 
and a ten-fold degree of ambiguity. 

On the other hand, a race of beings which had the 
greatest possible variety of expression would be able to 
speak with any degree of ambiguity at pleasure. It 
would have a distinct propositional form for restricting 
the universe to any one, one of any two, one of any 



66 ON THE ALGEBEA OF LOGIC. 

three, etc., of the possible cases ; or its entire number 
of propositions in two terms would be 2 16 or 2 16 1, 
according as one counts or does not count the case in 
which nothing is said. All the 65,536 or 65,535 things 
which can be said without using any other terms than 
theologians and scientists, for instance, the existing 
race is able to say, without very much difficulty, by 
combinations and alternations of its Aristotelian and 
Morganic propositions. To say that either no scientists 
are theologians (0, 2, 4, 6, 8, 10, 12, 14), or some theo 
logians are not scientists (3, 7, 11, 15), or some of those 
who are not theologians are scientists and some are not 
scientists (13), or else everybody is a theologian (1), is 
to make a statement of fourteen-fold ambiguity, to 
limit the constitution of the universe under considera 
tion to some case exclusive of 5 and 9. The contradic 
tory of a statement of the form 

(a v 6) + (a v &) + ( V 5) ( V^H @ v) 
is, by (13), 

(a V b) (a v b) (a v 5 + a y 5) (5 V) ; 

and to affirm that there are some theologians who are 
scientists, and that there are no theologians who are 
not scientists, and that either all scientists or else all 
non-scientists are theologians, and that not everybody 
is a theologian, is to affirm that either 5 or 9 furnishes 
the complete description of the universe with respect to 
the terms scientist and theologian. 

In three terms the number of combinations is 2 3 , the 
number of possible universes is 2 23 , = 256, and the num 
ber of possible propositions with all degrees of ambiguity 
is 2 256 . The types of universal propositions have been 
given by Mr. Jevons (XIII., p. 140), but the number is 
increased when single terms as well as combinations 



ON THE ALGEBRA OF LOGIC. 



67 



are permitted to be non-existent. Prof. Clifford s 
method for obtaining types (" Essays and Lectures. 
On the Types of Compound Statement involving Four 
Classes") is not difficult when applied to these terms. 
It takes account of terms which do not exist, and the 
number of types which he gives for four terms, 396, 
would be different on any other hypothesis. The prob 
lem would certainly be extremely difficult if such state 
ments as Mr. Jevons calls contradictory were excluded. 
Prof. Clifford s solution takes account of combinations 
only of universal propositions. The number of types 
of alternations only, and of alternations and combina 
tions of particular propositions only, is also 396, and 
the entire number is in this way raised to 4,396 ; but 
the determination of the number for mixed universal 
and particular propositions and for mixed alternations 
and combinations of them is still in the region of un 
solved problems. 

In three terms, the number of types of combinations 
of universal propositions is twenty-six, six four-fold, 
eight less than four-fold, and eight more than four-fold. 
The types of more than four-fold statement may be 
obtained by taking those combinations which are not 
excluded by the types of less than four-fold statement. 

LESS THAN FOUR-FOLD. 



I. 
II. 

III. 

IV. 
V. 

VI. 

VII. 

VIII. 





abc 

abc + abc 

abc + abc 

abc + abc 

abc + alJc + abc 

abc + abc + aBc 

abc + abc + abc 



68 



ON THE ALGEBBA OF LOGIC. 
FOUR-FOLD. 



IX. 

X. 

XI. 

XII. 

XIII. 

XIY. 



abc + abc + abc + abc 
abc + abc + abc + abc 
abc + abc + abc + abc 
abc + abc + aBc + abc 
abc + 5c + abc + 5c 
o&c + abc + a&c + a5c 



When condensed, these exclusions appear in the fol 
lowing form. The Arabic numbers give the correspond 
ing types in Mr. Jevons s Table. 



I. 


. . 





XXII. 


m 


1 


II. 


8 


abc 


XXI. 


. . 


a + b + c 


III. 


2 


ab 


XX. 


. . 


a + b 


IV. 


12 


(ab + al) c 


XIX. 


. . 


ab + ab + c 


V. 


11 


abc + abc 


XVIII. 


3 


ab + be + ca 


VI. 


7 


(a + b)c 


XVII. 


. . 


ab + c 


VII. 


9 


ab + abc 


XVI. 


4 


ab + ab + abc 


VIII. 


13 


abc + (ab + ab)c 


XV. 


15 


(a + b)c+(al + ab)c 



IX. 


10 


ab + be + ca 


X. 


. . 


a 


XI. 


5 


ac-\- be 


XII. 


1 


Obb + ab 


XIII. 
XIV. 


14 
6 


a (be + Ic) +a (lc + be) 
abc + (a + 5) c 



The exclusions 

IV., XVIII., XI., XII., XIII., 
are equivalent respectively to the identities 



XIV. 



ab = c. 



ON THE ALGEBRA OF LOGIC. 69 

In these Tables, the letters may represent propositions 
as well as terms ; of the 256 ways in which three propo 
sitions may be put together they give the 22 which are 
of distinct type. Case V., for instance, is the case in 
which three propositions, p l9 p^ p^ are affirmed to be not 
all at the same time true and not all at the same time 
false ; or, in other words, it is known that some one of 
them is true and some one of them is false. In case 
XVIL, p l and p 2 are not true together, and p z is not 
true at all. When the universe under discussion is the 
logical universe, the Tables serve to enumerate the 22 
possible types of argument between three propositions. 
In case IX., p ly p^ p s are propositions so related that 
from the truth of any one the falsity of the other two 
can be inferred ; in case XI., they are such that if two 
of them are both false or both true, the third is there 
fore false ; and, conversely, if that is false, the others 
are therefore either botli true or both false. The syllo- 
gism pip 2 pz v> i g ^ tne tyP e H- The argument "if 
either some animals covered with fur are black or some 
black things not covered with fur are animals, then 
some animals are black," that is, 

(abx V) + (abx v) \/(ab y), 

which is of the form (j?i +^ 2 )p 3 v> belongs to type 
VI. ; and the identity, 

(a \7 b) (c V d) ( a b + ca> \7)> 

belongs to type XIV. In order to find actual arguments 
of all the 22 types, it would probably be necessary to go 
into some hyper-universe where the laws of thought are 
different from those under which we reason. 



70 OX THE ALGEBRA OF LOGIC. 



BIBLIOGRAPHY. 

I. George Boole : The Mathematical Analysis of Logic, being an 
essay towards the Calculus of Deductive Reasoning. 1847. 

II. Boole : The Calculus of Logic. Cambridge and Dublin Math. 
Jour., Vol. III., 1848. 

III. Boole : An Investigation of the Laws of Thought, on which 
are founded the Mathematical Theories of Logic and Probabilities. 
1854. 

IV. Hermann Grassmann : Lehrbuch der Arithmetik. Berlin, 
1861. 

V. "W. S. Jevons : Pure Logic, or the Logic of Quality apart from 
Quantity. London and New York, 1864. 

VI. C. S. Peirce : On an Improvement in Boole s Calculus of 
Logic. Proc. Am. Acad. of Sciences, Vol. VI., 1867. 

VII. W. S. Jevons : The Substitution of Similars, the true Princi 
ple of Reasoning, derived from a modification of Aristotle s dictum. 
London, 1869. 

VIII. C. S. Peirce : Description of a Notation for the Logic of 
Relatives. Memoirs of the Am. Acad. of Sciences, Vol. IX., 1870. 

IX. Robert Grassmann : Die Formenlehre oder Mathematik. 
Zweites Buch : Die Begriffslehre oder Logik. Stettin, 1872. 

X. A. J. Ellis : On the Algebraical Analogues of Logical Rela 
tions. Proc. of the Royal Society of London, 1872-73. 

XI. Arthur Cayley : Note on the Calculus of Logic. Quart. Jour, 
of Math., Vol. XL 

XII. Ernst Schroder : Lehrbuch der Arithmetik und Algebra fur 
Lehrer und Studirende. I. Bd. : Die sieben algebraischen Opera- 
tionen. Leipzig, 1873. 

XIII. W. S. Jevons : Principles of Science. London, 1874. Third 
edition, 1879. 

XIV. Ernst Schroder : Der Operationskreis des Logikkalkuls. 
Leipzig, 1877. 

XV. J. Delboeuf : Logique Algorithmique. Liege et Bruxelles, 
1877. 

XVI. Hugh McColl : The Calculus of Equivalent Statements, and 
Integration Limits. Proc. London Math. Soc., Vol. IX., 1877-78. 
Vol. X., 1878, Vol. XL, 1880. 

XVII. Gottlob Frege : Begriffsschrift, eine der arithmetischen 
nachgebildete Formelsprache des reinen Denkens. Halle, 1879. Re- 



ON THE ALGEBKA OF LOGIC. 71 

viewed by Ernst Schroder in Zeitschrift fur MathematiJc und Physik, 
1880. 

XVIII. Wilhelm Wundt : Logik, eine Untersuchung der Princi- 
pien der Erkenntniss und der Methoden wissenschaftlicher Forschung. 
I. Bd. : Erkenntnisslehre. Stuttgart, 1880. 

XIX. J. Venn : On the Diagrammatic and Mechanical Repre- 
sentation of Propositions and Reasoning. Phil. Mag., July, 1880. 

XX. J. Venn : Symbolic Reasoning. Mind, July, 1880. 

XXI. C. S. Peirce : On the Algebra of Logic. Am. Jour, of Math., 
Vol. III., September, 1880. 

XXII. W. S. Jevons : Studies in Deductive Logic. London, 1880. 

XXIII. John Venn : Symbolic Logic. London, 1881. 

NOTE. In the foregoing article "combination" has been used as 
synonymous with "multiplication." In the following article, "combi 
nation " is used as including both multiplication and addition. 



ON A NEW ALGEBRA OF LOGIC. 

BY 0. H. MITCHELL. 



THE algebra of logic which I wish to propose may be 
briefly characterized as follows : All propositions cate 
gorical, hypothetical, or disjunctive are expressed as 
logical polynomials, and the rule of inference from a set 
of premises is : Take the logical product of the premises 
and erase the terms to be eliminated. No set of terms 
can be eliminated whose erasure would destroy an ag- 
gregant term. So far as the ordinary universal prem 
ises are concerned, the method will be seen to be simply 
the negative of Boole s method as modified by Schroder. 
The reason is, that the terms which the propositions 
involve are virtually all on the right-hand side of the 
copula, instead of all on the left-hand side, as in Boole s 
method. 

Attention is especially called to the treatment here 
given of particular propositions (of which there is intro 
duced a variety of new kinds) which is homogeneous 
with that of universals, the process of elimination being 
precisely the same in each case. For the sake of clear 
ness it may be well to state at the outset that I use 
addition in the modified Boolian sense, that is, x + y 
= all that is either x or y. 



ON A NEW ALGEBRA OF LOGIC. 73 

1. Logic has principally to do with the relations of 
objects of thought. A proposition is a statement of such 
a relation. The objects of thought, among which rela 
tions may be conceived to exist, include not only class 
terms but also propositions. The statement of a rela 
tion among propositions is a proposition about proposi 
tions, which Boole called a secondary proposition. But 
every proposition in its ultimate analysis expresses a 
relation among class terms. The universe of class terms, 
implied by every proposition or set of propositions, may 
be limited or unlimited. Two class terms, a, , are 
defined as the negatives of each other by the equations 

a + b = U, 
ab = 0, 

where U is the symbol for the universe of class terms. 
Two prepositional terms, a, /3, are defined as the nega 
tives or contradictories of each other by the equations 

a. + $ = GO, 

p=o, 

where oo is the symbol for the universe of relation, or 
for "the possible state of things." Mr. Peirce uses oo 
indifferently as a symbol for the universe of class terms, 
or for the universe of relation, but in the method of this 
paper it seems most convenient to have separate sym 
bols. We can speak of " all of " or " some of " 7, but 
hardly, it seems to me, of " all of " or " some of " the 
universe of relation ; that is, the state of things. For 
this reason oo seems an especially appropriate symbol 
for the universe of relation. 

The relation implied by a proposition may be con 
ceived as concerning " all of " or " some of " the uni 
verse of class terms. In the first case the proposition 



74 ON A NEW ALGEBRA OF LOGIC. 

is called universal ; in the second, particular. The rela 
tion may be conceived as permanent or as temporary ; 
that is, as lasting during the whole of a given quantity of 
time, limited or unlimited, the Universe of Time, or 
as lasting for only a (definite or indefinite) portion of it. 
A proposition may then be said to be universal or par 
ticular in time. The universe of relation is thus two- 
dimensional, so to speak ; that is, a relation exists among 
the objects in the universe of class terms during the 
universe of time. 

The ordinary propositions neglect the element of time ; 
and these will first occupy our attention. 

Let F be any logical polynomial involving class 
terms and their negatives, that is, any sum of products 
(aggregants) of such terms. Then the following are 
respectively the forms of the universal and the particular 
propositions : 

All Vis Fj here denoted by F 19 
Some CT is.*; F u . 

These two forms are so related that 



= oo 



that is, F l and F u are negatives of each other ; that is, 
\) = F u . The two propositions F l and J\ satisfy the 



one equation 



and are " contraries " of each other. Whence, by taking 
the negative of both sides, we get 

F U + F U = oo; 

that is, F u and F u are " sub-contraries " of each other. 
The line over the F in the above does not indicate the 
negative of the proposition, only the negative of the 



ON A NEW ALGEBRA OF LOGIC. 75 

predicate, F. The negative of the proposition F l is not 
jFi, but (^), which, according to the above, = F u . 

The Aristotelian propositions are represented in this 
notation as follows : 

(a + 5) x = All of U is + 5 = No a is 5, . . . . .#. 

(a#) M = Some of U is ab = Some a is b, .... /. 

(a + b\ = All of 7 is a + = All a is 5, .... A. 

(ab) u = Some of Z7 is a5 = Some a is not ft, . . 0. 

By substituting a, I for a, 5 throughout we get the four 
complementary propositions of De Morgan. If these 
two forms be applied to the sixteen possible sums of ab, 
aS, db, aS, there results the following 

TABLE OF PROPOSITIONS. 



(ab + al + ab + ab)^ .......... (0) 

(at + ab + 5) 1 ............ (ab) 

(ab + al + ab), ............ ( a 5) 

(ab + ab + ab) 1 ............ (ab) 

(ab + al + ab\ ............ (al) 

(ab + afyi ............ (5 + ab) 

(ab + ab^ ............ (a5 + al) 

(ab + al\ ............ yn + a fy 

(aB + ab\ ............ (al + a ^) 

(a5 + al\ ............ (ab + ab) 



............ (ab + ab + a^) u 

( al \ ............ (ab + ab + aJ). 

............ (a5 + ab + 05). 

............ (ab + a5 + ab) u 



76 



ON A NEW ALGEBRA OF LOGIC. 



Opposite propositions are negatives of each other. 
The Table reduced to its simplest form becomes 

REDUCED TABLE. 



1 


(U), 


(0) 


2 


(a + 5V 


(aft 


3 


(a + b], 


. lab) 


4 


(a + 5 s ), 


. (ab} u 


5 


(a + b), 


. (ab) u 


6 


(a), 


(a\ 


7 


(b} 


(b) 


8 




. . . (aB + ab) 


9 


(ob + db) l . . . 


. . . (ab + ab\ 


10 


(M 


(b) 


11 


(a\ 


(a} 


12 


(ab}, 


.(a + %) 


13 


(06} 


(a + b) 


14 


(ab\ 


. ( a 4. J) 


15 


(5)i - 


. (a + b), 


16 


(0)i 


(U}u 









If three terms be treated in a similar way we get 
2.2 23 ,= 512, different propositions. With n terms the 
total number is 2.2 2 ". 

The propositions (0)i and (0) tt assert that there is no 
universe of discourse, and are false in every argument, 
since a universe of class terms greater than zero is to be 
pre-supposed. Their negatives (7) M , (^)i are there 
fore true in every argument. The eight propositions of 
De Morgan occur in lines 2, 3, 4, 5 of the Table. 



ON A NEW ALGEBRA OF LOGIC. 77 

Since the universe of class terms is supposed greater 
than zero, the dictum de omni gives 

-*i-<^; 
that is, " all Z7is J 7 " implies " some U is I 7 ." 

To say " no UisF " is evidently the same as to say " all 

U is F;" that is, F = F 19 and since a proposition whose 
suffix is is thus expressible in a form with the suffix 
equal to 1, each suffix used will be supposed greater than 
zero. The suffix u in F u is taken to be a fraction or part 
of U less than the whole ; that is, " some of " U. In the 
proposition "some U is jP" it is not denied that all 

U may be J 7 , but the assertion is made of only a part 
of U. Thus u is taken as greater than zero and less 
than 1, or U. When u is written as a suffix of different 
propositions in the same argument, it is not meant that 
the same part of U is concerned in each case. F is writ 
ten for convenience instead of F v . Sometimes F e will 
be written as a form inclusive of both the forms F : and 
F u ; that is, e will be considered as having either of the 
two values 1 or u. 

For inference ~by combination of such propositions we 
have the following simple rules : 



The conclusion from the 
product of two premises is the 
product of the predicates of 
the premises affected by a suf 
fix equal to the product (in 
extension) of the suffices of 
the premises. Thus 



The conclusion from the 
sum of two premises is the 
sum of the predicates of the 
premises affected by a suffix 
equal to the sum (in intension) 
of the suffices of the premi 
ses. Thus 



* This is Mr. Peirce s sign for the copula of inclusion, being an abbrevia 
tion of ^. It is read "is," "is included under," or "implies." The 
following formulae are sometimes made use of in this paper : (a -< b) 
= (2> -< a) = (06 = 0) = ( oo = a -\- b), where a> = the universe of dis 
course. Also, (a -< b) (x -< y) -< (ax -< by). 



78 ON A NEW ALGEBRA OF LOGIC. 

When both premises are ( umver ^ al 1 the relation be- 

l particular 3 

tween the j P roduct 1 and the conclusion is equality ; 

I sum 
otherwise, the relation is -<, an implication. Thus 



(1) F& = (FG) 

(2) F,G u -< 

(3) F U G U -< oo. 



F U +G U = (F+G) U , (10 
F u +G,-< (F+G) U , (20 
F 1 +G 1 -<(F+G\.(3 ) 



These formula? are so evident as hardly to need explana 
tion. (1) means 

(U=F) (U=G) = (V-= 



and it follows from the definition of logical multiplica 
tion. By taking the negative of both sides, and chang 
ing F, & to F, G-, we get (! ) The law of the suffices 
in (! ) is u 4- u = u, or some + some = some. (2) means 



(U=F) (u=G) = (u = 

and follows also from multiplication. The law of suf 
fices is 1 u = u ; that is, Uu = u. Since G- 1 -< (7, (20 
follows from (! ). The law of the suffices is u + 1 = w ; 
that is, " all of " or " some of " = " some of," which is ad 
dition in an intensive sense. In formula (3) there can 
be no inference when nothing is known about the rela 
tion of the two suffices ; that is, F u G u <^ oo. If it be 
known that u and u have any common part, then for 
this common part F U G- U > -< (FG) UU ,. Thus if u = f Z7, 
and u =%U, then F u G- u , -< (FQ-\,, , where u" = uu =^U. 
Since we evidently have (FG-) U ^^ F U G U , we get by 
contraposition the formula (3 ), which means in words 
" either all U= F, or all U=Gr implies all U either 

Having regard to (1) and (I ), it will be seen that 



ON A NEW ALGEBRA OF LOGIC. 79 

the most general proposition under the given conditions 
is of the form 



or 



where F and G are any logical polynomials of class 
terms, II denotes a product, and S denotes a sum. 

If F and G be any of the sixteen polynomials involv 
ing two class terms a, 5, it is interesting to notice that 
any proposition, Z^IIG-J, can be reduced to the sum 
of products of the eight propositions of De Morgan. 
Thus, referring to the Table on page 76, any proposition 
F l in the first column is equivalent (1) to the product 
of one or more of the propositions 2,3, 4, 5, that is, E, 
A, E\ A (the two universal propositions added by De 
Morgan to the classic two being represented by E f y A ) ; 
and any proposition G u of the second column is equiva 
lent (I/) to the sum of one or more of the propositions 
7, (9, J , , the two accented letters representing the 
particular propositions added by De Morgan. Thus 
F l = U, and II G u = 112 /3 = 2U/3, where a is one of 
the four universals of De Morgan, and /3 is one of the 
four particulars. Thus 



Thus, for example, the proposition 

(a + b\ (ab + a5)j -f- (5) 1 + (&) u (# + ^)i> 
when reduced, becomes 

{(&) + (5) M + (&)} (a + &) 1 (a + 5) T 

4- ( + 5)! (a + 5) 1 (a + &) 1 + {()+ (a5) u }(a 

In like manner it may be shown that if F, G, etc. be 
logical functions of any number of class terms, a, 5, c, 
etc., the general proposition 



80 ON A NEW ALGEBRA OF LOGIC. 

may be reduced to a function of the eight propositions 
of De Morgan of the form 



where p, etc. are the eight propositions. 

Propositions united by + form disjunctive propo 
sitions. A hypothetical proposition, " if a, then /3," or 
a -< , where a and j3 are themselves propositions, is 
evidently equivalent to the purely disjunctive propo 
sition a + ft. Thus " if a is be, then cd is e " means 

(a + &<Oi-< + 5 + e)j; 
which is the same as 

(aB + ae) u + (e+S + e) 19 

which may be put into words in one way as follows: 
" some a is either non-6 or non-c, or all d which is non-c 
is e." The preceding formulas are examples of inference, 
by combination of propositions ; that is, of inference from 
a product or from a sum of propositions. 

Inference by elimination will now be considered. It 
will only be necessary to consider the fundamental form 
jP e , where e may be either 1 or u. If I 7 be a polynomial 
of the class terms, a,b,c,... x, y, z, then x, y, z may be 
eliminated from F e by erasure, provided no aggregant 
term is thereby destroyed. That is, 

Ft -< F, , 

where F 1 is what remains of F after the erasure. Thus 
(ax + bcxy + dcz + db) e -< (a + be + dc + db) e . 

The reason is obvious. To say that " (all or some) U 
is dx, or bcxy, or etc.," is saying by an obvious implica 
tion that " (all or some) U is a, or be, or etc." F e 
means (all or some) U -< F, and the erasure of a fac 
tor of a monomial term of F simply increases the extent 



ON A NEW ALGEBRA OF LOGIC. 81 

of the term ; therefore the predicate F is not diminished, 
and (all or some) U -<^ F , that is, FJ is a valid in 
ference. F 1 is really the sum of the coefficients of x, y, 
z in F, and is obviously a factor of F. The other factor 
of F is F + P ; for F (F + F ) = F, and F + F is seen 
to contain no factor independent of x, #, 2, since on 
erasing x, ?/, 2, the result is .F + F , = U. If one of the 
aggregant terms of F contain no letters except those to 
be eliminated, then its coefficient is U, and F t will in 
this case be a nugatory result. Thus from (a + led), 
b, c, d, be, Id, or cd can be eliminated, but not a, ab, ac, 
ad, ale, aid, aed, or led. As already stated, this alge 
bra is the negative of Boole s as modified by Schroder, 
so far as universal premises are concerned. Thus Boole 
multiplied propositions by addition, and eliminated by 
multiplying coefficients. The method here employed 
multiplies propositions by multiplication, and eliminates 
by adding coefficients. When many eliminations are 
demanded in a problem, the advantage in point of 
brevity of this method over Boole s is of course greatly 
increased. 

Before considering some illustrative examples, another 
kind of inference is to be noticed ; namely, inference by 
predication ; that is, the finding what a given proposition 
says about a given term, simple or complex. The rule 
is : Multiply F by the given term, m, or add m to F. The 
resulting coefficient of m in mF, or the residue of F after 
adding iii and reducing, will be the predicate of m. Thus 

F e -< (m = mF)., 
or F e ~< (m + F) t . 

The first means, "if U=F for all or some U, then 
m = mF for all or some U" and the result is obviously 
obtained by multiplying both sides of U= Fby m. The 



82 Otf A NEW ALGEBRA OF LOGIC. 

second relation means, "if U = F for all or some U, 
then U = m + F for all or some Z7," and the result is 
obtained by adding m to both sides, remembering that 
U + m = U. We have, of course, 

(m + F) e = (m + mF) e = (m = mF) e . 

I now give the solution of the well-known problem of 
Boole, " Laws of Thought," p. 146. The premises are, 
remembering that ( = &) = ( + b^ (a + 5)^ 

(x + z -f vyw + vwy) lf 

(v + x + w + yz + yz) 19 

(x + vy + w + wz\(xy + va; + w + wz)^ 

Multiplying the premises together, and dropping v from 

the result, we get 

(wxz + wxz -f- wfl32 + iZ?xy + wxyz) 19 = say J^. 
The four results asked for by the problem are 

(1) (x + wz + ws + 

(2) (wS + wz + ^ + w;y + 

(3) (y + ^^^ + wxz + wxz 

(4) (wa? + i/5S + xz + i 

The first gives the predicate of x in terms of #, ^, w, 
being the same as x -< wi + w^ + wjy, and is obtained by 
adding x to ^ and reducing. The second is the relation 
among y, 25, w, and is obtained by dropping x from F 
and reducing. The result (U\ shows that no relation 
is implied among ^, 2, alone. The third gives the 
predicate of y in terms of #, 2, w, and is obtained by 
adding # to .F and reducing. The fourth is the relation 
implied among x, z, w, and is obtained by dropping y 
from F and reducing. The relation (3) is not in its 
simplest form, since the implied relation (4) among #, 2, w 



ON A NEW ALGEBRA OF LOGIC. 83 

has not yet been taken into account. Since (p. 81) 
we have F = F (F + F 1 ) , where F ] is what remains 
of F after erasing ?/, and F + F is that factor of F 
which contains no factor independent of ?/, we get 
F l = F^F-i- F \. The first factor FJ is (4), and from 
the second factor we get (y + F+F \ as the simplest 
form of (3), that is, 

\ U i ^\ c * / 1 

Ordinary syllogism appears in this method as follows : 
The mood Barbara becomes 



b being eliminated by dropping it. The moods Darii, 
Datisi, Disamis, and Dimaris are all 

(ab) u (5+ c) x -< (abc) u -< (ac\. 
The premises of the mood Darapti are 

(m +p\(m + s) lf = (m + sp\\ 

but there is no conclusion independent of the middle 
term m, since m cannot be eliminated. In inferring the 
conclusion I from these two premises logicians have 
virtually included a third premise (w) tt , that is, " some 
of U is m," or " there is some ;." This with the pro 
duct of the other two gives " some 8 is p ; " that is, 

(m + sp) l (m) u -< (spm) u -< (sp) u . 

In the same way, the premises of Felapton and Fesapo 
are 

(m +p) 1 (m + 5) x = (m + sp)i> 

and m cannot be eliminated here. With the additional 
premise (m) u we get " some * is not p ; " that is, 

(m + sp\(m) u -< (spm) u -< (sp) H . 
The premises of Bramantip are 

(p + m)^ + s) 1 = (sp + sm + mp\ -< (s + p\\ 



84 ON A NEW ALGEBKA OF LOGIC. 

that is, the conclusion is not " some s is p" but " all p is 
," or " all 5 is p" the proposition A 1 . Here, again, the 
conclusion " some s is p " has been reached only by the 
virtual inclusion of a third premise, " there is some p" 
that is, (p) u . Then we have 

(sp + sm + mp)i(p)u ~< (smp) u ~< (sp) u > 

This is the same thing as to say that a particular con- 
elusion cannot be drawn from universal premises, since 
a particular proposition implies the existence of its sub 
ject, while a universal does not. 1 

As another illustration of the method, I solve the 
problem in Boole s "Laws of thought," p. 207. The 
premises are 

(w + xyz + xyz + xyz\, 



Their product is 

[f{wy + w (p~jr + pqr + pqr) + xyz + zyz + xyz (pTjr +pqr 
+ wx {pqyz + pqys + pqryz}\, say F 1 , 



which contains everything implied in the premises. The 
results asked for are 

1. (r + t + z) l9 whence t -< r + z 

2. (r+t + y) 19 " t-<r + y 

3. (U\ 9 

1 Mr. Peirce and others. 



ON A NEW ALGEBRA OF LOGIC. 



85 



(t + x) 19 
(p + <i + 



6. (t + yz + yzr) } 



7. (t -J- yz + y%)\ > 



whence 
tt 



whence 



whence 



t -< x 

y-<p + 2 

rt -< yz 
tz -< yr 
ry -< t 

t-< U 
,z + yz-< t. 



The relations in the first column are each obtained by 
dropping from F l the letters not concerned in the qurasi- 
tum. Each predicate in the second column is obtained 
by multiplying its opposite X" by its subject. The re 
sult 4 disagrees with that obtained by Boole. 

The two examples taken from Boole have dealt ex 
clusively with universal propositions. The following is 
of a different kind : 

What may be inferred independent of x and j from 
the two premises, " either some a that is x is not y, or all 
d is both x and y; " and "either some y is both b and x, 
or all x is either not y or c and not b " ? 

The premises are 

(axy) u + (d, + xy\, 
(bxy) u + (x + y + le\. 
By multiplication we get 

(axy) u (bxy\ + (bxy\ + (axy) u + (fix + ay + Icoi + lcxy) l . 
Whence, dropping x, y and reducing, we get 



which may be interpreted in words, " there is some 5, or 
a, or else all d is c and not 6." 



86 ON A NEW ALGEBRA OF LOGIC. 

From this result we may further eliminate c. Elimi 
nating c, we get 

(b + a) u + (3 + l) 19 

which means " either b or a exists, or no d is 5." 

The analogy bettveen class and propositioned terms. 
Hitherto in the consideration of F l and F u the polynomial 
F has been regarded as a function of class terms a, , 
etc. Suppose a, b, etc. to be prepositional terms like 
F l and F u , and call the resulting polynomial no longer F, 
but $. Then the suffices of ^ and M cannot be in 
terpreted any longer as referring to the universe of class 
terms, since the prepositional terms F 19 7^, etc., of which 
$ is a function, are supposed to have already suffices 
with this meaning. The suffices of $ x and $ M can only 
be interpreted then as referring to the universe of the 
time during which the complex or secondary proposition 
# is supposed to be true. Then, if F denote the uni 
verse of time, 

<>! means " $, during all F," or " all V -< <," 
$ v " "$, " some F," or "some F-< <3>." 
In otlier words 

^ means " $ is always true," 
< " " ^ is sometimes true," 
where " always " refers to the universe of time, V, 

Owing to the similarity between class terms and prepo 
sitional terms with respect to the operations of multipli 
cation and addition, it follows that all that has been 
said in regard to inference from propositions like F^ F u 
holds equally with ^ and $ v . Thus 



i 



ON A NEW ALGEBRA OF LOGIC. 87 

So in regard to elimination, any set of terms can be 
eliminated by neglect, provided no aggregant term is 
thereby destroyed. 

2. Propositions of Two Dimensions. 

Let U stand for the universe of class terms, as before, 
and let V represent the universe of time. Let I 7 be a 
polynomial function of class terms, a, 6, etc. Then let 
us consider the following system of six propositions : 

F uv , meaning "some part of U, during some part of V, is F" 
F ul) " " some part of U, during every part of V, is F" 
F lv , " "every part of U, during some part of V, is F? 
F ufl , " " the same part of U, during every part of F, is F" 
Fiv t " "every part of U, during the same part of V, is F" 
F n , " "every part of U, during every part of F, is F." 

By thus introducing the element of time, three varieties 
of the proposition F u are distinguished, F uv , F ul , F U>1 . 
Thus in speaking of the people of a certain village during 
a certain summer (Z7= village, V= summer), "some 
of the Browns were at the sea-shore during the sum 
mer" may mean either that some of them were there 
during a part of the summer, or that some of them were 
there during every part of the summer, not necessarily 
the same persons, or that the same persons werepthere 
during the whole summer. These three meanings are 
here denoted respectively by (&)? ()i> (&s)i- Three 
varieties of F 1 are also distinguished, F n , F lv , F lv ,. 
Thus " all the Browns were ill during the year " may 
mean either that every one was ill during every part of 
the year, or that every one was ill during some part of 
the year, not necessarily the same part, or that every 



ON A NEW ALGEBKA OF LOGIC. 

one was ill during the same part of the year. These 
three meanings are denoted respectively by (I + i] 
P + ^P + fV 

The dictum de omni gives the following relations 
among these six propositions : 



ul ,F lv F ul F uv) and F^ + 
and since same is included under some, we have 

F lv ,-< F ln and F^ -< F ul . 
The following pairs of propositions, 

F uv and F u ,F ul and F* , F U>1 and F l9 , 
satisfy the two equations 



and the members of each pair are therefore the negatives 
or contradictories of each other. Thus if F= I +\ it is 
seen that (U\ v and (5 + i) u are contradictories ; that is, 
" either some of the Browns were not ill during some 
part of the year, or they were all ill during every part 
of the year," and both cannot be true. An example 
of the second pair is (li) ul and (5 + ?) ll/; that is, " either 
some of the Browns were ill during every part of the 
year (not necessarily the same persons during the whole 
year) or at some particular time none of them were 
ill," a*nd both cannot be true. An example of the third 
pair is (fo ) ttl and (b + i) lv , " either the same Browns 
were ill during the whole year, or it was true for each 

* The natural first thought is that FU, F u i, FI V , F uv form a system of 
propositions by themselves, but it is seen that FW and F u >i must be added 
to the system, in order to contradict F u} and F lv . Mr. Peirce pointed out 
to me that these propositions are really triple relatives, and are therefore 
six in number. F n , for instance, means "F is a description of U during 
V" See the Johns Hopkins University Circular, August, 1882, p. 204. 



ON A NEW ALGEBRA OF LOGIC. 89 

part of the village during some part of the year that none 
of the Browns were ill," and both cannot be true. 

Since from A -< B we get A + B = GO and AB = 0, 
so from F n -<^ F w 



we get F vv + F uv = GO, 

and ^ii^ii = 0; 

hence F n and "F n are " contraries " of each other, and 
F uv , F uv are " sub-contraries." In the same way F& -< F lv 

gives F vl + F lv = GO, 

and F lv ,F v >! = ; 

that is, F w and F^ are contraries, and F vl , F lv are sub- 
contraries. The line over F affects only F, not the 
suffices. Thus the negative of F n would be written 
(3y, not F n . 

To say " no U is ^, during F" is evidently the same 
as to say " all Uis F, during F; " that is, 



so F 10 F u , 



Since every proposition with zero as one or both of the 
suffices is thus expressible in a form with no suffix equal 
to zero, each suffix used will be supposed greater than 
zero. The suffices u, v are also supposed less than U, F", 
just as u was supposed less than U in the preceding 
section. F aji will sometimes be used to include all six 
of the fundamental propositions : that is, a will be con 
sidered as having any one of the values 1, w, or u ; ft as 
having any one of the values 1, v, or v . 



90 



ON A NEW ALGEBKA OF LOGIC. 



For inference ly combination of such propositions we 
have tup following simple rules, which are seen to be the 
same as in 1 : 



The conclusion from the 
product of two premises is the 
product of their predicates 
affected by suffices which are 
the products (in extension) of 
the suffices of the premises. 
Thus 



When all the suffices are 1. 
the relation between the pro 
duct of the premises and the 
conclusion is equality ; other 
wise it is -<, that is, impli 
cation. Thus 



F ul G n -<(FG) ulJ 
etc. 



The conclusion from the 
sum of two premises is the 
sum of their predicates affected 
by suffices which are the sums 
(in intension) of the suffices 
of the premises. Thus 



When none of the suffices are 
1, the relation between the 
sum of the premises and the 
conclusion is equality ; other 
wise it is -<, that is, impli 
cation. Thus 

F uv + G UV =(F + G) m , 

F lv + G uv -< (F+ G) uv , 

etc. 



But, by an exception to the 

rule, do not have F ul G lv -< (FG\ V since G lv is not of the 

form (Gj) v . 

These formulas really follow at once from those in 1. 
Thus F n may be written (^1)1 ; hence by 1 we have 



G) 



and 

(F u ) v + (G u ) v = (F u + G u ) v =((F+ 
So in general we have 



and 



(F a + 



O1S: A NEW ALGEBRA OF LOGIC. 91 

the addition of the suffices being taken in the same sense 
as in 1 ; that is, 

1 + 1 = 1, l + u = u , 

1 + U U, U + U 1 = Uj 

u + u u, u + u u j 

with like equations for v, v . The second set of equa 

tions means 

All of + same part of = same part of, 

Some of + same part of = some part of, 
Same part of -f- same part of same part of, 

and a -little consideration will show that the formulae 
hold as well for the accented suffices as for the unac 
cented. 

The following formula is evident : 

(FG)afi ~< Fo.fi Ga.fi 

For inference by elimination we have only to consider 
the general form F a ^ and the rule is precisely the same 
as the rule for elimination given in 1, viz. : Any set of 
terms may be eliminated by erasure provided no aggregant 
term is thereby destroyed. Thus 

(a + bx + cdxy + ey) a fi -< (a + b + cd + e) a fi, 

and the reason of the rule need not be repeated. 

The rule for inference by predication is also evidently 
the same as that previously given. Thus 

(a + b + cd + e) a p -< (c -< a + b + 
and, in general, 



If, after the multiplication has been performed, mF= mP, 

then we have 

F a -<(m-< P) a/3 . 



92 ON A NEW ALGEBRA OF LOGIC. 

Since propositions of the form F n can be multiplied 
without loss of content, and propositions of the form F uv 
can be added without loss of content, the most general 
proposition involving the six fundamental elements is of 
the form 



or H (S^ u + 3G ul + SJGT^ + 2K lv + 2L lv + M uv ), 

where F, 6r, etc. are logical polynomials of class terms. 
But to the six elements just considered we may add as 
elements the forms $ x , <2> r considered at the close of 1, 
where $ is of the form P u + 2 Q^ or PJIQ U (see page 
79) ; so that $ 1? <& v will be of the forms 

(P u +2Qi)i, WI ft,).. 

It is clear that (P M + SQ^ V = P uv + 5Q lv , and that 
(^P i nQ u ) l = P n nQ ul ; but for the two forms of $ M <P V 
just given, no such reduction can be made. The suffices 
within the parentheses of ^ 1 , $ v refer to the universe of 
class terms, those outside to the universe of time. If 
the relative meaning of these suffices be reversed, so that 
the suffices inside the parentheses refer to the universe 
of class terms and those outside to the universe of time, 
we have two other prepositional elements. Thus in 
order to distinguish the meaning of the suffices clearly, 
it will be necessary to use the capital letters Z7, F", and 
write the four forms just considered as 



or, in full, 

(P u + 2Q v )r, (P^nft,),, (p v 



The negative of <fr y is $ r , which is of the form *. So the 
negative of X v is X u , which is of the form fl u . As ex 
amples of r , X^, suppose the universe of class terms 
to be plane figures a, 6, etc., on a blackboard, and the 



ON A NEW ALGEBRA OF LOGIC. 93 

universe of time to be an hour. Let P = cib, and 
Q = c + d ; then 



means " during every part of the hour, either some a 
is b, or no c is d" while 



means " for every part of the blackboard, it is true that 
it is either sometimes both a and b, or never both c 
and d" So, as examples of " and fl u we have, re 
spectively, 

\(ab)oQ + &)} 

which means " at some time during the hour, all the 
blackboard is a&, and some of it is c + d" and 



which means " some part of the blackboard is always 
ab and sometimes c + c?." 

Adding the four prepositional elements just described 
to the six described previously, we see that the most 
general proposition is of the form 



To illustrate the method of inference from propositions 
like the foregoing, consider the solution of the following 
problem : 

Six plane figures, a, b, c, d, e, f, on a blackboard are 
constantly changing their size, shape, and position during 
an hour under the following restrictions : 

I. The area of c and d together is always included in 
the area of a and b together, or else, during a certain 
portion of the hour, e is equal to the part common to d 
and f. 



94 ON A NEW ALGEBKA OF LOGIC. 

II. The part of a which is not e is always included 
under the part common to d and f which is not b, or else, 
during the whole hour, it is true for some part of the 
board that all b is both c and e. 

III. Either a and d are non-existent and e always 
covers the board, or else it is always covered either by b 
or by c. 

What may be inferred (1) about the relation among 
a, c, e and f, independent of b and d; (2) about the re 
lation among a, c, e, independent of b, d, f ? 

The premises are 

I. (a + b + cd) n + (def + de + ef\ v , 

II. (d + e + ldf) n + (5 + ce) ul 

III. ade 



From the product of the first two we infer 

(ab + acd + ae + be + cde + a5df) n + (def + ade + aej) 
+ (al + led + ace + bce) ul + (tdef+ Me + 



and multiplying this proposition by the third premise 
according to the preceding rules, we get as an inference 

(Me + acae) n + (abode -\- dbcde) ul + (bdef + abcle + abef) w 
+ (cdef+ acde + acef) lv , + (ale + ace + bce) ul + (cdef+ Me 
+ lcej) uv + {(ab + be) u + (abc + ace + bee + alcdf) L } y , 

three of the complex elements reducing to simple ones 
according to the formulae, 



G) uv . 
Dropping b and d from the above proposition, we get 

(ae) n + (ae) ul + (ae + ef) w + (ace + cef) lv , + (ac + ce) ul 
+ (ce + cf) w + {(a + e) v + (ac + ce + cf)^}^ 



ON A NEW ALGEBRA OF LOGIC. 95 

But in a sum, any term may be dropped which implies, 
or is included under, another term. 

O)n -< (e) Bl and (ace + cef) lv , -< (ae + ef) w ; 
therefore the above reduces to 

(ae) ul + (ae + ef) lv , + (ac + ce) ul + (ce + cf) m + {(a + e) L 
+ (ac + ce + cf)v}r, 

which is the first quaesitnm, and may be read in words 
" either it is always true that some e is not a ; or at a 
particular part of the hour all a is e, and all e is /; or 
during each part of the hour some c is either a or e ; 
or at some part of the hour some c is either/ or not e; 
or during each part of the hour either all a is e, or the 
whole blackboard is c and all a is either e or/." 
Dropping / from this result, we get 

(a)i + ( + e) lv , + (ac + ce) ul +(c) uv + {(a + e) n + (c) L } y . 

But (ac + ce) ul -< (c\ v and {(a + e) n + (c>)^-< (a + e) w 
(<?), therefore we get as the second qua3situm, 



+ 



which means " either it is always true that some e is not 
a ; or during some particular part of the hour all a is e ; 
or there is sometimes some c." In like manner any 
other set of terms can be eliminated by dropping them 
from the product of the premises. 

Propositions of more than two dimensions. If the 
universe of relation be supposed to consist of three di 
mensions, 7", V 9 W, proceeding just as before we should 
find that the number of fundamental propositions with 
three suffices, 

^in > ^uio Fa.> F uvw , etc., 
is twenty-six. The logic of such propositions is a " hyper " 



96 ON" A NEW ALGEBRA OF LOGIC. 

logic, somewhat analogous to the geometry of "hyper" 
space. In the same way the logic of a universe of rela 
tion of four or more dimensions could be considered. 
The rules of inference would be exactly similar to those 
already given. 

Allusion has already been made to the fact that the 
propositions considered in this and the preceding section 
may be regarded as relative terms. In the first section, 
the two fundamental propositions, F l and F u , are dual 
relatives. F t means "F is a description of every part of 
U-, " and F u means "F is a description of some part 
of 7." Thus F l and F u correspond to the two funda 
mental dual relatives. So in 2, F n is a triple relative 
term, meaning 11 F is a description of every part of U 
during every part of V" Thus the six fundamental 
propositions of two dimensions correspond exactly to the 
six fundamental varieties of triple relatives, and so on. 

3. On Certain Other Methods. 

The propositions A and in Mr. Peirce s notation 
are, respectively, 



Y, 
X^< Y. 

Mr. McColl expresses them in a similar way, using a 
different symbol for the copula. Both Mr. McColl and 
Mr. Peirce have given algebraic methods in logic, in 
which the terms of these propositions are allowed to 
remain on both sides of the copula. 

In the method of 1 (of which 2 is an extension), 
the propositions A and are expressed as follows : 

(X+ Y) 19 equivalent to oo -< X + Y, 



ON A. NEW ALGEBRA OF LOGIC. 97 

that is, all the terms of the universal proposition are 
transposed to the right hand side of the copula, while 
those of the particular proposition are transposed to the 
left-hand side. 

If these propositions be expressed in the reverse way, 
namely, 



the rules of inference become the exact logical negatives 
of those in 1, addition taking the place of multiplica 
tion, arid vice versa. XY-<^ is equivalent to (XY) , 
meaning "none of U is XY" as has already been ex 
plained. GO ^<; X + .Fmay be represented by (X+ Y) q , 
meaning " some of U is not X + Y" or " there is some 
thing besides X+ Y." Thus jP and F q are the two 
fundamental forms of proposition in this method, arid 
the rules of inference by combination are 



F G =(F+G\ 
F,G q - < (F + G) < 



F+ G = 



F q + G Q -< (FG) q 
F + G -< (FG\. 



Elimination is performed by multiplying together the 
co-efficients of the quantities to be eliminated. 

Boole s method, as simplified by Schroder, lias been 
extended by Miss Ladd, in the foregoing paper, so as to 
express particular propositions without the use of Boole s 
objectionable " arbitrary " class symbol. She has ex 
pressed A and as follows : 

XY V, equivalent to XY -< 0, 

XY\[, " " XY^Q. 

Thus F and F u are the two fundamental forms of propo- 



98 ON A NEW ALGEBKA OF LOGIC. 

sition in her method, and the rules of inference ly 
combination are 



F Q G U -<(FG) U 

F u G u -< oo. 

Elimination from F is performed by multiplying co 
efficients ; from jP M , by adding them. 

One more method remains to be noticed, the negative 
of Miss Ladd s method, in which A and are expressed 

as 

o>-< X+ Y, 



and where F 1 and F q are thus the two fundamental forms 
of proposition. The rules of inference ly combination are 

F q +Q q =(FGT) q 

(F+G) q F Q +G,-<(FG\ 



oo 



and elimination from F l is performed by addition of co 
efficients ; from F q , by multiplication of coefficients. 

4. On a special notation for De Morgan s Eight Propo 
sitions, with an extension of the same to similar propo 
sitions of three or more terms. 

It is proposed in this section so to change the notation 
previously given for De Morgan s eight propositions that 
the elimination of the middle term will be performed by 
an algebraic multiplication of the premises. Denote by 
J , E , , A what J, E, 0, A become when each term 
is replaced by its negative. The propositions J, E, 0, A, 
and their complementaries I , E , , A have already 
been represented (see page 76) respectively by 






OK A NEW ALGEBRA OF LOGIC. 99 



and also, since F l F Q , by 

(ab) u , (5)o, (a5), (a&) , 
Let these be now chaned to 






where the negative of a term is now denoted by affecting 
it with the exponent 1, and the negative of a propo 
sition is denoted in the same way. Thus 

(ab~ l ) means " some a is not b" 
(ab- 1 )- 1 " "all a is 6," etc. 
With this notation there is the following simple 

RULE OP INFERENCE. Excluding products of two par 
ticulars, the conclusion from a set of premises is their 
algebraic product, with the convention that the appearance 
of a middle term in the result indicates that there is no 
conclusion. 

Thus, Barbara is 

- 1 )- 1 x (sm- 1 )- 1 -< (sp- l )~\ 



and Darii is 

(mp~ l )- l x (sm)-< (sp)-, 

but from A and as premises we get 

(rap- 1 )" 1 X (sra- 1 ) -< oo, 

the middle term not disappearing from the product. 

From the nature of this notation, just as with that of 
1, the order in which the two terms of a proposition are 
written is indifferent, and consequently the figure of a 
syllogism -is indifferent. Thus, (mp) is the same as 
(pm). Thus Celarent and Cesar e are 

(mp)~ l x (sm- 1 )- 1 -< (sp)~\ 
Darii and Datisi are 

(mp- l )~ l X (sm) -< (sp). 



100 



ON A NEW ALGEBRA OF LOGIC. 



Ferio, Festino, Ferison, and Fresison are 
(mp)- 1 X (sm) -< (sp- 1 ). 
Camestres and Camenes are 
- 1 X 



Baroko is 



(pm~ l )- } X 



1 -< (sp)-\ 
-< (sp- 1 . 



Bokardo is 



Disamis and Dimaris are 

X (ras" 1 )" 1 -< (sp). 

X (ms- 1 )- 1 -< (sp- 1 ). 

This rule of inference is seen to accord with the now- 
recognized invalidity of the moods Darapti, Felapton, and 
Fesapo. Thus the premises of Darapti are 



(mp~ l )~ l x 

from the product of which m does not disappear, and 
there is therefore, according to the rule, no inference. 
The same is true for Felapton and Fesapo. The premi 
ses of Bramantip are 

(pm~ 1 )- 1 X (ms~ l )~ l , which -< (s" 1 ^)" 1 . 

The following Table gives all the valid moods from 
De Morgan s eight propositions : 

- l m^) (prn) 



(sm) 1 
(sm- 1 ) 
(s-%) 



(sm) 



(sp) 



(9) 






to) 



ON A NEW ALGEBRA OF LOGIC. 101 

There are twenty-four valid moods, but if no distinction 
be made between s and p, these reduce to the twelve in 
either half of the Table, the Table dividing itself sym 
metrically along the diagonal from left down to right. 
The unsymmetry of the Aristotelian system is seen from 
the fact that the fifteen valid moods of the Aristotelian 
system comprise only eight out of the twenty-four of the 
Table, and these eight select themselves very unsym- 
metrically, being those underscored by dotted lines. 
From the three formulae 



~ l -< (sp)-\ 
(sm) X (pm)- 1 -< (sp~ l ), 
(sm)~ l X (pm) -< (s-, 

the whole twenty-four syllogisms of the Table may be 
obtained by substituting for m, s, and p their negatives 
in all possible ways, each formula yielding eight. 

Mr. Hugh McColl, in his papers on logic in the " Pro 
ceedings of the London Mathematical Society " (Vol. IX, 
et. seq.), has been using a notation for the copula identi 
cal in meaning with that of Mr. Peirce. He uses a colon 
to denote implication, instead of -<. Mr. Peirce has 
recently told me that Mr. McColl justifies his use of the 
colon by its mathematical meaning as a sign of division. 
Thus Barbara and Celarent are 

m : p m : p 

s : m s : m 

. . s :p /. s :p, 

and the analogy to division is obvious. But this analogy 



102 ON A NEW ALGEBRA OF LOGIC. 

exists only in the two universal moods of the first figure. 
Thus Cesare and Festino are 

p : m p : m 

s:m s -7- m 

.-. s :p s+p, 

where -f- is the negative copula, and the analogy to 
division is wanting. In the notation of this section the 
analogy of the premises to ratios, and of the conclusion 
to their product is more nearly complete. 

Extension of the preceding. 

Let (abc) denote " a, b, c have something in common," 

and (abc)~ l " a, b, c " nothing " " 

By substituting for a, 5, c their negatives in all possible 
ways, we get sixteen propositions concerning three terms, 
thus seen to be analogous to De Morgan s eight concern 
ing two terms. In the same way we may get thirty-two 
propositions concerning four terms, and 2.2 n propositions 
concerning n terms. The formulas of inference from 
propositions like the above are 

(ab...ffh...t) (h...lm...q)~ l -< (ab...g) (m...q)~\ 
(ab Id) (l- l m q)~ l -< (ab km...q)~ l . 

In the first, where one premise is particular, inference 
can take place independently of any number of middle 
terms, provided each term is positive in both premises, 
or negative in both. In the second formula, when both 
premises are universal, inference can take place inde 
pendently of only one middle term, and this must be of 
different quality in the two premises. By an obvious sub 
stitution these two formulae are reduced to the formulae 



ON A NEW ALGEBRA OF LOGIC. 103 

previously given involving only two terms in each pre 
mise. Thus 



(r 1 *)- 1 -< (a*)- 1 . 

That is, the premises of the first mean " that which is 
common (x) to a, 5, ...#, has something in common 
with the common part (#) of A, . . . Z ; " and " the common 
part (j/) of h, . . . I has nothing in common with m, . . . q" 
Whence the inference is (xy~ l ), or (ab . . .g) (m. . . q)~ l . 
The premises of the second mean " whatever may be 
common (x) to , b, ... Jc, has nothing in common with 
Z; " and " whatever may be common (z) to m, . . . q, has 
nothing in common with non-Z." Whence the inference 
is (xz)~ l , or (ab . . . km . . . q)~\ 

(abc) means (ab) (ac) (be), 
/. (abc) 1 " (ab)- 1 + (ac)- 1 + (bc)~\ 

Thus any one of these propositions is reducible to a 
function of De Morgan s eight. 

5. Note on De Morgan s Twenty Propositions. 1 

It is proposed in this section to consider a simple 
method of deriving and writing De Morgan s Twenty 
Propositions. Let A = all of A, a part of A, A = all 
of non-J., and d = part of non-J., where part of is under 
stood to mean less than the whole of. Let a second term 
B be modified in the same way. Then, by affirming 
and denying identity between each modification of the 
first term and each modification of the second, we get 
thirty-two propositions, of which, however, twelve are 
duplicates. That is, the process yields twenty distinct 

1 See his "Syllabus of Logic," 24-62. 



104 



ON A NEW ALGEBRA OF LOGIC. 



propositions, and they are easily seen to be the twenty 
of De Morgan. Let the affirmation of identity between 
two terms be denoted by their juxtaposition, and let the 
denial of the same be denoted by a line extending over 
both terms. Then we have the following 

TABLE OF DE MORGAN S TWENTY PROPOSITIONS. 



AB, or AB 


AB, or AB 


AB, or AB 


AB, or AB 


Ab, " aB 


Ab, aB 


~Ab, " ^ 


Ab, " aB 


aB, " Jb 


aB, " AB 


aB, " Tb 


~OB, " A5 


ab 


ab 


ab 


~ab 


al 


al 


db 


a5 



Thus, AB means " the whole of A is identical with the 
whole of j." It is obvious that AB is equivalent in 
meaning to AB. The second proposition, Ab, means 
" the whole of A is identical with a part of B" (that is, 
all_ A is B, and some B is not A). It is clear that 
aB, or " a part of non-A is identical with the whole of 
non-,5," is the same as Ab. To take an example from 
the other side of the Table, AB means " it is not true 
that the whole of A is identical with the whole of B." 
This is simply the denial of the proposition AB. Ab 
means " it is riot true that the whole of A is identical 
with a part of 5," a simple denial of Al. 

The propositions below the horizontal line of division, 
which are differentiated from those above the line by 
containing only small letters in their symbols, are De 
Morgan s eight " simple " propositions. 



ON A NEW ALGEBRA OF LOGIC. 105 

(A part of A is a part of B } 

~ (Some A is B > 

( It is not true that a part of A is a part of B \ 

~~ I ~vr A 7? i 

- _ ( A part of A is a part of non-B ) Q 

\ Some A is not B ) 

-Y ( It is not true that a part of A is a part of non-B ) , 
= lAll.iis2? > 

The remaining four of these eight are derived from these 
four by the negation of their terms. This notation for 
the eight propositions differs only slightly from that 
employed in previous sections. 

De Morgan derived his eight " simple " propositions 
by applying the Aristotelian forms A,E,I, to the four 
pairs of terms X, Y\ X,Y\ X,Y; X, Y. This process 
gives sixteen propositions, of which eight are duplicates. 
The other twelve of the twenty he called "complex," 
because they are compounded of the eight simple propo 
sitions, as follows : 



~al)Xtib AB = al + ab 

Ab aft X ab Ab ab + ab 

aB = aL X ab a,B = aB + ab 

~AB d6 X ab AB = at + ab 

~Ab ~ab X ab Ab = ab + ab 

aB = ab X ab ~aH = at + ab 

The following Table gives the conclusions from one 
hundred out of the possible four hundred combinations 
of two premises from this system of twenty propositions : 



106 ON A NEW ALGEBEA OF LOGIC. 

PM Pm pM pm pm PM Pm pM pm pm 



SM 

Sm 
sM 
sm 
sm 
SM 
S^ 
^M 
sm 
sm 


SP 


sP 


Sp 


sp 


sp 


SP 


sP 


Sp 


sp 


sp 


Sp 


sp 


Sp 





sp 











Sp 


sp 


sP 


sP 


sp 


sp 














sp 


sP 


sp 





sp 








o 








sp 





sp 


sp 




















o 


sp 


SP 


























o 


Sp 
7P 











o 





o 


o 








o 





o 











o 








sp 


sP 


sp 


sp 








o 


o 





sp 


w 


sp 


Sp 





sp 











sp 






By applying the sign of negation first to the S, then to 
the P, then to both the S and the P, the remaining three 
hundred are obtained. According to De Morgan, who 
postulates that every term and its negative is greater 
than zero, there are two conclusions not given in the 
Table, namely : 

sm X pin -< sp, 

sm X pm -< sp, 

and from these are obtained six others by applying the 
sign of negation to s and p. But according to the 
definitions of Mr. Peirce and others, already alluded to, 
these are invalid conclusions ; since, being particular, 
they imply the existence of their subjects, while the 
universal premises do not. 



OPERATIONS IN RELATIVE NUMBER WITH 
APPLICATIONS TO THE THEORY OF PROBA 
BILITIES. 



BY B. I. OILMAN. 



THE purpose of this Paper is to deduce the formulae 
for the addition and multiplication of Relative Number, 
and to apply them in demonstrating the well-known 
fundamental theorems of Probabilities, according to Mr. 
Peirce s method of dealing with the subject. 

If a relation be that which we perceive when a group 
of objects are viewed together, but which we do not 
perceive when we regard each separately, then any act 
of comparison will bring to view a relation. If the 
objects compared are two in number, the relation may 
be called a dual one. 

Such a dual relation may be viewed in two lights, or 
we may say it splits into two elementary forms, accord 
ing as one or the other object is our starting-point in 
comparing the couple. The two are called the direct 
relation and its converse. Thus, what is ordinarily 
termed a relation may be said to have ends, being based 
on a comparison having a direction. One of these ends 
is called the relate, the other the correlate. 

A relative number is a number obtained in either of 
the two following ways : first, by dividing the number 



108 OPERATIONS IN EELATIYE NUMBER. 

of instances in which a given relation has a relate in a 
certain class of objects by the number of objects in the 
class ; or, second, by dividing the number of instances 
in which a given relation has a correlate in the given 
class by the number of objects in the class. Hence, for 
a given relation p we have two such relative or aver 
age numbers, one, the number of instances in which 
p has a relate of the class ?/, divided by the number 
of ?/ s ; and the other, the average number per y of p 
whose correlates are ?/ s. The former might be called 
the relate-number of p , the latter its correlate-number. 
But if we extend the class y to include all the objects 
in the universe, since the number of instances in which 
the relation p occurs having a relate which is an object 
in the universe, is equal to the total number of times 
p occurs at all, and the same thing is true of the number 
of occurrences in which it has a correlate which is in the 
universe : it follows that for both relate and correlate 
numbers we get the average number of relations p per 
object in the universe. That is, any relation p has but 
one (what we shall call) general relative number. 

Denoting each object in the universe by a certain 
letter, each possible different couple of objects (con 
sidering those couples as different in which the same 
elements occur in a different order) will be symbolized 
once, and only once, in Mr. Peirce s scheme of pairs, 
as follows: 

A:A A:B A:C A:D ... 

B:A B:B B:C B:D ... 

C:A C:B C:C C:D ... 

D:A D:B D:C D:D 



OPERATIONS IN RELATIVE NUMBER, 109 

Now if in this scheme of pairs we assume the relation- 
direction to be constant, say from left to right, that 
is, that the right-hand members of the pairs are the cor 
relates, it will follow that any single instance of any 
relation must subsist between some one, and only one, 
of the pairs. Marking in any way, as by a circum 
scribed circle, those pairs between the components of 
which subsists the relation p ; and marking by a circum 
scribed square instances of the relation p", we shall 
have in general some pairs surrounded by circles, some 
by squares, and some by both. 

Whence if p and p" denote respectively the number 
of individual relations comprised in the general relations 
p and p", we shall have 

p -f p" = number of pairs surrounded by circle alone + num 
ber of pairs surrounded by square alone + twice 
the number of pairs surrounded by both circle and 
square = p ,p" + P ",p + 2 p , P " 

in which p , p" denotes the number of pairs concerning 
each of which it can be said that it is in both the rela 
tions p and p" ; and p , p" denotes the number of pairs 
which arc . at once in the relation p 1 and not in the 
relation p". Again, 

p -f p" = number of pairs in circle, or square, or both + num 
ber in both = (p 1 -I- p") + p 1 , p", 

in which according to Mr. Jevons s notation (p 1 -|-p") 
denotes that class of pairs concerning each member of 
which it can be said that it is either an instance of p or 
of p" or of both. Now, since a general relative number 
is the total number of individual instances of a relation, 
divided by the number of objects in the universe, if we 
indicate the number of objects in the universe by oo, 
-^ will indicate the general relative number of the rela- 



110 OPERATIONS IN RELATIVE NUMBER. 



tion p r . Symbolizing this quotient by [y ] , and dividing 
both sides of the above equations by oo, we have 



[p ] + [P"] = IP , P"-] + [A p ] + 2 |y, p"] = [y .,. p /] + [y, p"]. 

We thus have reached two formulae for the addition 
of two relative numbers. Similarly, we have for the 
addition of three relative numbers 



] + [P"] + [> "] - CP , ? , P "] + [P", P , P "] + [P ", P ,P"] 
+ 2 [ P , p", p" ] + 2 |y , p ", p//] + 2 [p /, p ", p/] 



or 

- [p ! P" -i- P /;/ ] + CP , P", P "] + iy, p" f , P"] 

+ [p^P // ,P / ] + 2[ P ,p^^/], 

Similar formulae may be deduced for the addition of n 
relative numbers, as follows : 



P n - 2 ]) 
P re - 3 ]) 



or 

= [p -|-p"-|Y" - ... .j.p] 

+ LP>p">p" ?"]+...+ [p"" 1 ^*?? p l ~ 2 ] 



This latter formula gives, when the relations are 
mutually incompatible, that is, when no two of them 
can subsist between the same pair, a much simpler 
result : 



CP ; ] + [P"] + . . . + [p"] = [p .|.p".|.p " . . 
all the other terms reducing to zero. 



OPERATIONS IN RELATIVE NUMBER. Ill 

To obtain a formula for the multiplication of relative, 
numbers we notice that 



Let x, which may be any number, signify the number 
of different existing groups of three objects, such that 
the first is to the second in the relation p and the second 
to the third in the relation p". Such a group may be 
called a relative sequence, and may be denoted by p r p" 
without the comma. Then 



If now 

the formula becomes 



In this case, therefore, the product of the relative 
numbers of the two given relations equals the relative 
number of the sequence formed from them. 

Multiplying numerator and denominator of by the 

00 * 

number of objects in the universe, it becomes p X 2 . 

The numerator of this fraction is a number equal to the 
number of different triplets obtained by combining each 
p with every object in the universe. Between the second 
and third members of these triplets either the relation 
p or p 1 must hold ; and no relative sequence of the form 
p p" or p p n can exist which does not appear among them. 
Hence the number p x oo equals the sum of the num 
bers of p p" and p p". The denominator being the square 
of the number of objects in the universe is equal to the 



112 OPERATIONS IN RELATIVE NUMBER. 

number of possible pairs, and each of these is either 
p n or p 



00 
and 



or 

I 

P" 




P P" _ P P" 



That is, the average number of sequences p f p n per 
each p" is the same as the average number of sequences 
p p n per each p f . Hence, whether the relations in which 
any given individual stands to the others in the uni 
verse are all p", or one or more p" and the rest p n , will 
make no difference on the average in the number of 
relative sequences whose first member is p of which it 
is the intermediary. The number of such sequences in 
the case of any individual being the number of the ob 
jects standing to it in the relation p multiplied by the 
number of objects in the universe, it follows that the 
number of objects standing to any given individual in 
the relation p is not affected by the circumstance of its 
being p" to one or more objects. 

P P" P j. P P" P" i 

Similarly, from ^- = we may get *--~- = , whence 

eZ = p p " + pp or ^ = ; that is, whether an object 
P p+p P p 

is correlate in any relations p or not, will make no 
difference on the average in the number of p"s of which 
it is the relate. 

For instance, letting p indicate the relation borrower 
from, and p" the relation trustee of, this condition ex 
presses, first, the fact that a man s being a trustee makes 
no difference on the average in the number of borrowers 



OPERATIONS IN" RELATIVE NUMBER. 113 

from him ; and, second, that a man s being a lender or 
not makes no difference on the average in the number 
of funds which he controls as trustee. Such relations, 
from one of which nothing can be inferred regarding 
the presence of the other, are called independent re 
lations. Hence for independent relations, 

!> ] x 



The expression p p" here denoting the number of 
relative sequences of that form, if we define a compound 
relation to be a combination of such relative sequences 
as have the same individual object as relate, , and also 
the same individual object as correlate, f/ , we shall have 
each compound relation consisting of as many sequences 
as it has intermediary objects. Hence, in order to ex 
press the number of p p f $ in terms of compound relations 
of that form, to the total number of compound rela 
tions we shall have to add the number of those which 
have two intermediaries, since they each contribute an 
extra sequence ; and to this sum we must further add 
twice the number of compound relations having three in 
termediaries, three times those having four, etc. Hence 
we have for the number of relative sequences expressed 
in terms of compound relations, 



2 



= P P" + P P" + 2 P P" + ...( n -l) P P 



o^ 

wherein P P" denotes the total number of compound 
relations of the form p p 1 having whatever number of 
intermediaries; P P" denotes the number of such com 
pound relations having two intermediaries, etc. Whence, 
dividing through by oo, we have 

... (n - 



114 OPERATIONS IN RELATIVE NUMBER. 

and the following formula results for the multiplication 
of independent relative numbers : 

- 2 3 

CP ] [P"] = [^P"j + \_PiP"-] + 2 [P P//] . . . ( n - 1) [PP"]. 
By a somewhat different and a longer process of proof, 
it can be shown that for independent relations the follow 
ing formula holds for the multiplication of n relative 
numbers : 



3 

2[P . . . 



Here it is to be noted that the superscribed numbers do 
not refer to the number of intermediaries, but to the de 
gree of connection, the number of ways in which relate 
and correlate n are connected by chains of relation. 

The continued product of the numbers indicating the 
simultaneous intermediaries at the successive steps, it 
is easily seen, cannot be less than r nor greater than 
r (n ~ l \ when the connection in the given relation is an 
r-fold one. Since permuting the multipliers does not 
change the left-hand member, the right-hand member 
remains constant in whatever order the elementary rela 
tives are compounded. 

Through the addition formula we have reached what 
we may call polynomial relative numbers, of the form 
[p ! -I- p" -I ..... |. p M ] which expresses the relative number 
of that class of pairs, each one of which is an instance 
of some one or more of the relations p . . . p n . In the 
case of incompatible relations we have the equation 

[p -lV -l ..... |-P B ] 



Whence the multiplication of polynomial relative num 
bers reduces in the case of incompatible relations to that 
of monomials. 



OPERATIONS IN RELATIVE NUMBER. 115 

The involution of a monomial relative number gives 
the ordinary result of multiplication, except that all the 
elements of the resulting compound relation are the 
same. If we involve an incompatible polynomial, we 
shall get a result according to the multinomial theorem, 
consisting of monomial powers and products. 

In order to apply these results to the theory of proba 
bilities, we shall require to make a supposition in regard 
to the character of the relations we are to consider. If 
a relation is perceived whenever we compare objects, it 
follows that a relation will be noticed when we think 
of an object as existing at successive times; for this 
involves a comparison between its aspect at one time 
and at another. 

This relation between objects which differ, so far as 
we see, only in existing at different times, we call iden 
tity. The pairs in the principal diagonal of the relative 
scheme exist in this relation only, since what we call the 
same or an identical object is both correlate and relate. 

The relative number of the relation of identity is evi 
dently unity, since it occurs once, and no more, for every 
individual in the universe. Now we can, if we please, 
agree to bring the various individual relations, that is, 
relations subsisting between individual objects, which 
together make up the total extension of the general re 
lation identity, into various classes according to the 
character of the objects they identify. This will create 
as many kinds of relation of identity as there are classes 
of objects in the universe, and their relative numbers will 
vary from - up to unity, and will express the propor 
tion of objects of the different kinds in the universe. 

Further, we may agree to take for the divisor of our 
relative number, for our y, instead of all the objects in 
the universe, some limited portion of them, say the class 



116 OPERATIONS IN RELATIVE NUMBER. 

b. This will be a return to the special relative number 
mentioned at the beginning of the paper ; but it is evi 
dent that since the relation whose relative number we 
seek is a relation of identity, every instance of it which 
has its relate in the class b will also have its correlate 
in that class, and vice versa; so that the relate and cor 
relate number of the relation will be the same, and may 
be called simply its relative number. Such a relative 
number will mean the number of identity relations of 
the form a to be found among the relations pertaining 
to the individuals of the class b divided by the number 
of those individuals; that. is, the number of a s among 
the 6 s, divided by the number of & s, or, in other words, 
the proportion of the genus b that is of the species a. 
If we regard events as the objects between which the 
relations we are considering subsist, an identical relative 
number will express the proportion in which a certain 
species of event exists in a genus. With this ratio will 
vary the expectation with which we shall look to see a 
case of the genus a case also of the species ; it may be 
said to measure the value of the genus as a proof of the 
species, to measure, that is, the prove-ability, or proba 
bility, of the species from the standpoint of the genus. 

On this view of probability it has to do, not with 
individual events, but with classes of events ; and not 
with one class, but with a pair of classes, the one 
containing, the other contained. The latter being the 
one with which we are principally concerned, we speak, 
by an ellipsis, of its probability without mentioning the 
containing class; but in reality probability is a ratio, 
and to define it we must have both correlates given. 

An identical relative number, then, when the identities 
considered are events, will be the ratio of a specific to a 
generic occurrence ; and this ratio is called the proba- 



OPERATIONS IN RELATIVE NUMBER. 117 

bility of the species with respect to the genus. The 
mathematical combination of probabilities will therefore 
take place in accordance with the formula for relative 
number already reached, with such modifications as re 
sult from their application to relations of identity. 

In establishing by these formula the fundamental the 
orems of probabilities, let the individuals in the uni 
verse we are considering be events ; and let a denote a 
certain kind of relation of identity between them, that 
is, a certain class of events, and a the remaining rela 
tions of identity, that is, all the rest of the events in the 
universe. The general relative numbers of a and a 
that is, the general probabilities of a and a in the uni 
verse will be denoted by [a] and [#]. 

From the addition formula we have 

[>] + [] = [a.j.a] + [>,] 

The first term of the right-hand member is the relative 
number of that class of pairs, each of which exhibits 
either or both of the relations a and a ; and the second 
term of the right-hand member is the relative number 
of that class of pairs, each of which exhibits both the 
relations a and a. But since by definition a is a part 
and a the rest of the existing relations of identity, no 
event exhibits them both, and [a ,a] ; while the num 
ber of relations a .|. a equals oo, and hence [a .\. a] = 1. 
Thus we have 

M + []=! 

[5] =!-[>] (1) 

or, the probability of the negative of an event equals 
unity minus the probability of the event. 

The relations a and a are incompatible relations ; that 
is, they cannot subsist at once between the same pair. 

Incompatibility means, therefore, in the case of rela- 



118 OPERATIONS IN RELATIVE NUMBER. 

tions of identity between events, that no one event can 
be of both species ; the species are mutually exclusive, 
- the events, as we say, cannot happen together. Such 
events may be called exclusives, and we may denote 
by the term alternatives specific events which together 
make up a genus; that is, exclusives one or other of 
which must happen if the generic event happen at all. 
The generic event consisting of the occurrence of any 
one of a number of exclusives may be called an alter 
nating event. 

The abridged form of the addition formula, when the 
relations are incompatible, gives the following as the 
probability of an alternating event : 

[> -I- * -I- <H I- "] = M + p] + [c] + . . . + W (2) 

That is, the probability of an alternating event is equal 
to the sum of the probabilities of the exclusives of which 
it is composed. 

The expression a , b ,c , cl . . . n denotes an event which is 
at once a, 5,0, note?... and not n\ and [a,b,c,d. . .n~\ 
denotes the probability of such a compound event. If we 
have certain events of known probability, a,b,c . . . n 
which are not exclusives, and wish to obtain the proba 
bility of the occurrence of some one, and only one, of 
them, the desired expression reduces to a sum of such 
compound probabilities. For the event in question will 
be either (a, 5, . . . w,w), or (a, 6 . . . m,w), etc., 
or (a, 5 . . . m,ri) ; and these compounds being mutu 
ally exclusive, the event is an alternating one, and its 
probability is expressed as follows : 

[a,5 . . . n.\.a,b . . . n.\. . ..\.a . . . m,n~\ = 

[a,l . . . n] + \_a,b . . . ri] + . . . + [a . . . m,w] 

This result being in terms of the probability of compound 



OPERATIONS IN RELATIVE NUMBER. 119 

events, to make it available we must have means of cal 
culating compound probabilities from simple ones. 

The formula obtained above for multiplying relative 
numbers expresses the result of such a multiplication in 
terms of the relative numbers of compound relations. 
In the case of identical relations, these would be com 
pound relations of identity. But since no object or 
event is in the relation of identity to more than one ob 
ject or event, that is, itself, each compound relation 
of identity must consist of a single relative sequence; 
accordingly all the terms after the first in the right-hand 
member of the multiplication formula disappear, the re 
maining term being the relative number of a relation 
of identity compounded of all the multiplied factors. 
But since all the objects concerned in this compound 
relation from relate to correlate n are one and the same, 
it is no longer a sequence of relations, but a coexistence 
of special identities, a coexistence of characters ; and 
its relative number is the relative number of such co 
existences, of objects or events in which coexist all 
the given special identities that belong at once to all the 
given species. The condition that the relations should 
be independent, that is, that between any two of them, 

a, b a,T) 

T T 

for relations of identity becomes the condition that the 
proportion of 5 s that are also a s should equal the pro 
portion of 5 s that are also o s ; in other words, that an 
event is b should make it neither more nor less likely 
that it is also a case of a, and vice versa. 

We thus see that the multiplication of identical rela 
tive numbers, when the relations are independent, will 
give the relative number of the events in which all the 
multiplied identities coexist. The probability of a com- 



120 OPERATIONS IN RELATIVE NUMBER. 

pound event, therefore, when the components are inde 
pendent, may be found by multiplying together the prob 
abilities of all the components. Applying this principle 
to the case of the compound events 

[a, I ... u] + \_a,b, ... n\ ... + [... m,n\, 
we have for the probability of the occurrence of one, and 
only one, of n independent non-exclusive events, 
[a, I . . .n.\.a,b,c . . . n.\ ..... \.a . . . m,ri] 
= M P] [>][>] + [a] [6]...[w]...+ []...[m] [>]. (3) 
For the probability of the occurrence of some one or 
more of n independent non-exclusive events, we obtain 
by transposition from the second form of the general 
addition formula, 

& -I- * ! c -I ..... |- n] = [a] + [] + . . . . + [>] 
- H M P] . . [n\ -...-[]... [m] [n] 
-2[a]J?][c][<Z]...[n]-. . .-2[]...[q [m] [ w ] 

-(-!) M[ft][c]. . . M (4) 

Since the probability of a compound event is the pro 
duct of the probabilities of the components (when inde 
pendent), we have the following equation : 

[a,M...n]== M [&] [c] . . . M (5) 

which gives us 



or 



that is, the probability of any event is equal to the 
probability of any compound event into which it enters, 
divided by the probability of the compound event made 
up of the remaining components. 



OPERATIONS IN RELATIVE NUMBER. 121 

"We may obtain an expression for the probability of 
a compound event when the components are not inde 
pendent, by noticing that in establishing the formula 
for multiplication the independence of the relations ena 
bled us to substitute in the left-hand member of the 

equation, -^ for ^-. If the relations are not independ 
ent, this is not permissible ; whence indicating ^~- by 
\_p p"] P " the equation reads 



ov >[>"] 

or for identical relations 

[,] [] = [,], 

in which [&,6] & denotes the proportion of a, 6 s among 
5 s, the probability that an event of the genus b will also 
be of the species a. An extension of these considera 
tions gives the general formula 

[>,&.. .w] 6 ... B [&,c.ra] c ... [c,d..ri] dtttn ...\m 9 n] n [ri] = 

O,a...rc]; (6) 

that is, the probability of a compound event, when the 
components are not independent, is equal to the general 
probability of any one of the components multiplied by 
the probability that one of the other components will 
happen when the first happens, and so on until all the 
components are exhausted. 

Let us suppose that the compound event, instead of 
being composed of n different events, is composed of 
n like events, a. If these different occurrences of a are 
independent, that is, if the fact that a has occurred 
once, makes it neither more nor less likely that it will 
occur again, we have 

M = W (7) 



122 OPERATIONS IN RELATIVE NUMBER. 

While the mere fact that a has occurred will not, 
contrary to the popular notion, make it any more or less 
likely to recur, it is evident that in many instances at 
tendant circumstances, as in the case of habit, may de 
stroy the independence of successive occurrences. 

If a is a compound of independent relations of identity, 
as &, 6, c, ... m, the formula becomes 

[(a,ft,c. m ) M ] = KM- - m T 

= (M P] M - - - W)- 

= MP][c]...[m]; (8) 

that is, the probability of the repetition of a compound 
event n times is equal to the product of the n ih powers of 
the probabilities of its components. 

We have seen that a polynomial relative number ex 
presses the probability of the occurrence of some one 
or more of the separate events symbolized therein. If 
the events are exclusives, it expresses the probability of 
the occurrence of some one of them. 

Considering two exclusives, a and 5, in order to ob 
tain the probability that one or other of them should 
occur n times, it is to be noticed first that this event 
itself is not a single compound event, but a compound 
alternating event, consisting of as many compound alter 
natives as there are different arrangements of a and I in 
n occurrences. Since the probability of an alternating 
event is the sum of the probabilities of the alternatives, 
the probability we seek will be the sum of the probabili 
ties of all the compound alternatives ; that is, the sum 
of all the products obtained by forming all possible 
arrangements of n simple probabilities, each of which 
must be either [a] or [b~\. In other words, the opera 
tion of finding the probability of the occurrence of one dr 
other of two exclusives n times, is the same as that of 



OPERATIONS IN RELATIVE NUMBER. 123 

raising the binomial [a] + [5] to the n th power. This 
is otherwise seen thus : Since a and b are exclusives, 



but 



Similarly, for more than two exclusives, the probability 
of one or other happening p times is equal to the sum 
of the probabilities of the exclusives raised to the p^ 
power, or 

[(<H-H<H ..... I-")*] = (M + P] + W + M) p . (9) 
It may be observed in relation to the probabilities of 
the compound alternatives of which these sums are made 
up, that any one will be equal to all the others in which 
the elementary exclusives enter in the same proportions, 
although in different orders. The case of highest proba 
bility will evidently be that consisting entirely of that 
one of the elementary exclusives which has the highest 
probability, and the case of lowest probability will be 
that in which the elementary exclusive having the lowest 
probability alone appears. On the contrary, other con 
siderations show that the most probable proportions in 
which different alternatives will enter into a series of 
trials will be the ratios of their probabilities, while the 
most improbable proportions will be those exhibited by 
series consisting entirely of some one of the alternatives. 
The same thing is true of exclusives ; the most probable 
proportion in which they will be found in a series of 
trials being the ratios of their probabilities. But while 
with alternatives the sum of the probabilities of all 
possible orders will continue to be unity, however the 
number of trials is increased, with exclusives the sum 
of these probabilities will decrease in geometrical pro 
gression as the trials are repeated. 



124 OPERATIONS IN RELATIVE NUMBER. 

The results thus far reached, readily lead to other com 
binations of probabilities, as in the following examples : 
The probability of the occurrence of at least one of two 
events with a third is given by the equation 



(10) 

in which, as in general in probabilities, the events are 
supposed to be independent. 

When a and b are exclusives, the same probability is 
equal to 

([a] + [J]) []. 

For any number of exclusives, and any number of 
other events, the equation becomes 

[OH-0-1 ..... 1-"), a, ft, . . n\ = 



For the probability of the occurrence of one, and only 
one, of any number of non-exclusive events with any 
number of others, we have 

[(a,/3 . . . v .|. . .|. ... .|.a, . . . p.,v)a,b. . . n] = 

[a] [J] . . . W( [a] [ft... [v]+ . . . + [a] . . . [][>]) (12) 

The probability that a will occur m times to n occur 
rences of 5, that is, that m a s will happen while n 6 s 
are happening, will be the probability of the compound 
event consisting of m a s and n 6 s. The probability 
that m a s will be succeeded by n 6 s is [a] m [] n , and the 
number of different arrangements of m + n objects, m 

of one kind and n of another, is , , ; whence the total 

\m [n 

probability is 



If a and b were alternating events, this expression 
would give the probability of the occurrence of some one 



OPERATIONS IN RELATIVE NUMBER. 125 

or other of TT exclusives m times, while some one or 
other of p exclusives is happening n times. Substituting 
the values assumed in this case by [a] m and [6J n ,we 
have for this probability 



+[/?]+...+ wr (M +[] + !>])" a*) 

In this investigation of some modes of combining 
probabilities, suggested by the consideration of Relative 
Number, we have used the Addition formula in reaching 
(1) the probability of negative events, (2) of some one 
of n exclusives, (3) of some one, and only one, of n 
non-exclusives, and (4) of at least one of n non-exclu- 
sives. From the Multiplication formula we have ob 
tained the probability of a compound event when the 
components are either (5) independent, or (6) depend 
ent ; and by a reference to the involution of Relative 
Number have established formula for the probability 
of the repetition of (7) simple (8) compound or (9) 
alternating events. These results have been combined 
in the more complicated cases (10 - 14) last considered. 



A THEOEY OF PROBABLE INFERENCE. 

BY C. S. PEIRCE. 



I. 

THE following is an example of the simplest kind of 
probable inference : 

About two per cent of persons wounded in the liver recover ; 
This man has been wounded in the liver; 
Therefore, there are two chances out of a hundred that he 
will recover. 

Compare this with the simplest of syllogisms, say the 
following : 

Every man dies ; 

Enoch was a man ; 

Hence, Enoch must have died. 

The latter argument consists in the application of a 
general rule to a particular case. The former applies to 
a particular case a rule not absolutely universal, but sub 
ject to a known proportion of exceptions. Both may 
alike be termed deductions, because they bring informa 
tion about the uniform or usual course of things to bear 
upon the solution of special questions ; and the probable 
argument may approximate indefinitely to demonstration 
as the ratio named in the first premise approaches to 
unity or to zero. 



A THEORY OF PROBABLE INFERENCE. 127 

Let us set forth the general formula of the two kinds 
of inference in the manner of formal logic. 

FORM I. 
Singular Syllogism in Barbara,. 

Every M is a P; 
S is an M ; 
Hence, S is a P. 

FORM II. 
Simple Probable Deduction. 

The proportion p of the M a are P s ; 

S is an M ; 

It follows, with probability p, that S is a P. 

It is to be observed that the ratio p need not be exactly 
specified. We may reason from the premise that not 
more than two per cent of persons wounded in the liver 
recover, or from " not less than a certain proportion of 
the JTs are P s," or from " no very large nor very 
small proportion, etc." In short, p is subject to every 
kind of indeterminacy; it simply excludes some ratios 
and admits the possibility of the rest. 

The analogy between syllogism and what is here called 
probable deduction is certainly genuine and important ; 
yet how wide the differences between the two modes of 
inference are, will appear from the following considera 
tions : 

1. The logic of probability is related to ordinary syllo 
gistic as the quantitative to the qualitative branch of the 
same science. Necessary syllogism recognizes only the 
inclusion or non-inclusion of one class under another; 
but probable inference takes account of the proportion 



128 A THEORY OF PROBABLE INFERENCE. 

of one class which is contained under a second. It is 
like the distinction between protective geometry, which 
asks whether points coincide or not, and metric geome 
try, which determines their distances. 

2. For the existence of ordinary syllogism, all that is 
requisite is that we should be able to say, in some sense, 
that one term is contained in another, or that one object 
stands to a second in one of those relations : " better 
than," " equivalent to," etc., which are termed transitive 
because if A is in any such relation to B, and B is in 
the same relation to (7, then A is in that relation to 0. 
The universe might be all so fluid and variable that 
nothing should preserve its individual identity, and that 
no measurement should be conceivable ; and still one 
portion might remain inclosed within a second, itself 
inclosed within a third, so that a syllogism would be 
possible. But probable inference could not be made in 
such a universe, because no signification would attach to 
the words " quantitative ratio." For that there must be 
counting ; and consequently units must exist, preserving 
their identity and variously grouped together. 

3. A cardinal distinction between the two kinds of 
inference is, that in demonstrative reasoning the con 
clusion follows from the existence of the objective facts 
laid down in the premises ; while in probable reasoning 
these facts in themselves do not even render the con 
clusion probable, but account has to be taken of various 
subjective circumstances, of the manner in which the 
premises have been obtained, of there being no counter 
vailing considerations, etc. ; in short, good faith and hon 
esty are essential to good logic in probable reasoning. 

When the partial rule that the proposition p of the 
M 9 s are P s is applied to show with probability p that 
8 is a P, it is requisite, not merely that S should le an 



A THEORY OF PROBABLE INFERENCE. 129 

Jlf, but also that it should be an instance drawn at ran 
dom from among the M a. Thus, there being four aces 
in a picquet pack of thirty-two cards, the chance is one 
eighth that a given card not looked at is an ace ; but 
this is only on the supposition that the card has been 
drawn at random from the whole pack. If, for instance, 
it had been drawn from the cards discarded by the 
players at piquet or euchre, the probability would be 
quite different. The instance must be drawn at ran 
dom. Here is a maxim of conduct. The volition of 
the reasoner (using what machinery it may) has to 
choose S so that it shall be an M\ but he ought to 
restrain himself from all further preference, and not 
allow his will to act in any way that might tend to 
settle what particular M is taken, but should leave that 
to the operation of chance. Willing and wishing, like 
other operations of the mind, are general and imperfectly 
determinate. I wish for a horse, for some particular 
kind of horse perhaps, but not usually for any individual 
one. I will to act in a way of which I have a general 
conception ; but so long as my action conforms to that 
general description, how it is further determined I do 
not care. Now in choosing the instance 8, the general 
intention (including the whole plan of action) should 
be to select an M, but beyond that there should be no 
preference ; and the act of choice should be such that if 
it were repeated many enough times with the same in 
tention, the result would be that among the totality of 
selections the different sorts of M s would occur with 
the same relative frequencies as in experiences in which 
volition does not intermeddle at all. In cases in which 
it is found difficult thus to restrain the will by a direct 
effort, the apparatus of games of chance, a lottery- 
wheel, a roulette, cards, or dice, may be called to our 



130 A THEORY OF PROBABLE INFERENCE. 

aid. Usually, however, in making a simple probable 
deduction, we take that instance in which we happen at 
the time to be interested. In such a case, it is our 
interest that fulfils the function of an apparatus for 
random selection ; and no better need be desired, so 
long as we have reason to deem the premise " the pro 
portion p of the M s are P s " to be equally true in 
regard to that part of the M s which are alone likely 
ever to excite our interest. 

Nor is it a matter of indifference in what manner the 
other premise has been obtained. A card being drawn 
at random from a picquet pack, the chance is one-eighth 
that it is an ace, if we have no other knowledge of it. 
But after we have looked at the card, we can no longer 
reason in that way. That the conclusion must be drawn 
in advance of any other knowledge on the subject is 
a rule that, however elementary, will be found in the 
sequel to have great importance. 

4. The conclusions of the two modes of inference like 
wise differ. One is necessary ; the other only probable. 
Locke, in the "Essay concerning Human Understanding," 
hints at the correct analysis of the nature of probability. 
After remarking that the mathematician positively knows 
that the sum of the three angles of a triangle is equal to 
two right angles because he apprehends the geometrical 
proof, he then continues : " But another man who never 
took the pains to observe the demonstration, hearing a 
mathematician, a man of credit, affirm the three angles 
of a triangle to be equal to two right ones, assents to it ; 
that is, receives it for true. In which case, the founda 
tion of his assent is the probability of the thing, the proof 
being such as, for the most part, carries truth with it ; 
the man on whose testimony he receives it not being wont 
to affirm anything contrary to or besides his knowledge, 



A THEOEY OF PROBABLE INFERENCE. 131 

especially in matters of this kind." Those who know 
Locke are accustomed to look for more meaning in his 
words than appears at first glance. There is an allusion 
in this passage to the fact that a probable argument is 
always regarded as belonging to a genus of arguments. 
Tliis is, in fact, true of any kind of argument. For the 
belief expressed by the conclusion is determined or caused 
by the belief expressed by the premises. There is, there 
fore, some general rule according to which the one suc 
ceeds the other. But, further, the reasoner is conscious 
of there being such a rule, for otherwise he would not 
know he was reasoning, and could exercise no attention 
or control ; and to such an involuntary operation the 
name reasoning is very properly not applied. In all 
cases, then, we are conscious that our inference belongs 
to a general class of logical forms, although we are not 
necessarily able to describe the general class. The dif 
ference between necessary and probable reasoning is that 
in the one case we conceive that such facts as are ex 
pressed by the premises are never, in the whole range of 
possibility, true, without another fact, related to them as 
our conclusion is to our premises, being true likewise ; 
while in the other case we merely conceive that, in rea 
soning as we do, we are following a general maxim that 
will usually lead us to the truth. 

So long as there are exceptions to the rule that all 
men wounded in the liver die, it docs not necessarily 
follow that because a given man is wounded in the liver 
he cannot recover. Still, we know that if we were to 
reason in that way, we should be following a mode of 
inference which would only lead us wrong, in the long 
run, once in fifty times ; and this is what we mean when 
we say that the probability is one out of fifty that the 
man will recover. To say, then, that a proposition has 



132 A THEORY OF PROBABLE INFERENCE. 

the probability p means that to infer it to be true would 
be to follow an argument such as would carry truth with 
it in the ratio of frequency p. 

It is plainly useful that we should have a stronger 
feeling of confidence about a sort of inference which will 
oftener lead us to the truth than about an inference that 
will less often prove right, and such a sensation we do 
have. The celebrated law of Fechner is, that as the 
force acting upon an organ of sense increases in geo 
metrical progression, the intensity of the sensation in 
creases in arithmetical progression. In this case the 
odds (that is, the ratio of the chances in favor of a 
conclusion to the chances against it) take the place of 
the exciting cause, while the sensation itself is the feel 
ing of confidence. When two arguments tend to the 
same conclusion, our confidence in the latter is equal to 
the sum of what the two arguments separately would 
produce ; the odds are the product of the odds in favor 
of the two arguments separately. When the value of the 
odds reduces to unity, our confidence is null ; when the 
odds are less than unity, we have more or less confidence 
in the negative of the conclusion. 

II. 

The principle of probable deduction still applies when 
S, instead of being a single Jtf, is a set of M s, n 
in number. The reasoning then takes the following 
form : 

FORM III. 
Complex Probable Deduction. 

Among all sets of n M s, the proportion q consist each of 
m P s and of n m not-P s ; 



A THEOEY OF PROBABLE INFERENCE. 133 

S, S r , S ff , etc. form a set of n objects drawn at random 
from among the M*s : 

Hence, the probability is q that among S, S , S", etc. there 
are m P s and n m not-P s. 

In saying that S, S , S", etc. form a set drawn at ran 
dom, we here mean that not only are the different in 
dividuals drawn at random, but also that they are so 
drawn that the qualities which may belong to one have 
no influence upon the selection of any other. In other 
words, the individual drawings are independent, and the 
set as a whole is taken at random from among all possi 
ble sets of n M s. In strictness, this supposes that the 
same individual may be drawn several times in the same 
set, although if the number of M s is large compared 
with n, it makes no appreciable difference whether this 
is the case or not. 

The following formula expresses the proportion, among 
all sets of n M s, of those which consist of m P s and 
n m not-P s. The letter r denotes the proportion of 
P s among the M s, and the sign of admiration is used 
to express the continued product of all integer numbers 
from 1 to the number after which it is placed. Thus, 
4 ! = 1 . 2 . 3 . 4 = 24, etc. The formula is 

r" (1 r )n-m 

q = nl X : X 

m I (n m) I 

As an example, let us assume the proportion r = f 
and the number of M s in a set n = 15. Then the 
values of the probability q for different numbers, m, of 
P s, are fractions having for their common denominator 
14,348,907, and for their numerators as follows : 



134 A THE GET OF PEOBABLE INFERENCE. 



TO 


Numerator of q. 





1 


1 


30 


2 


420 


3 


3640 


4 


21840 


5 


96096 


6 


320320 


7 


823680 



m 


1\ umerator of q. 


8 


1667360 


9 


2562560 


10 


3075072 


11 


2795520 


12 


1863680 


13 


860160 


14 


122880 


15 


32768 



A very little mathematics would suffice to show that, 
r and n being fixed, q always reaches its maximum value 
with that value of m that is next less than (n + 1)?*,* and 
that q is very small unless m has nearly this value. 

Upon these facts is based another form of inference to 
which I give the name of statistical deduction. Its gen 
eral formula is as follows : 

FORM IV. 

{Statistical Deduction. 
The proportion r of the M ? s are P ? s ; 

S f , ;/ , S>", etc., are a numerous set, taken at random 
from among the Hf s : 

Hence, probably and approximately, the proportion r of 
the S s are P s. 

As an example, take this : 

A little more than half of all human births are males ; 
Hence, probably a little over half of all the births in New 
York during any one year are males. 

We have now no longer to deal with a mere probable 
inference, but with a probable approximate inference. 

* In case (n-f- l)ris a whole number, q has equal valut-s for m = 
(TI -|- 1) r and for m (n -f- 1) r 1. 



A THEORY OF PROBABLE INFERENCE. 135 

This conception is a somewhat complicated one, meaning 
that the probability is greater according as the limits of 
approximation are wider, conformably to the mathemati 
cal expression for the values of q. 

This conclusion has no meaning at all unless there be 
more than one instance ; and it has hardly any meaning 
unless the instances are somewhat numerous. When 
this is the case, there is a more convenient way of ob 
taining (not exactly, but quite near enough for all practi 
cal purposes) either a single value of q or the sum of 
successive values from m = m\ to m = m 2 inclusive. The 
rule is first to calculate two quantities which may con 
veniently be called ^ and t. 2 according to these form 
ula : 



^ ffll _ (n _j_ 1) r 
tl = y2nr(l rj 



_ 



1 -f m 2 (n + 1) r 



where w 2 > Wi- Either or both the quantities ^ and ^ 
may be negative. Next with each of these quantities 
enter the table below, and take out | 9^ and | 9t 2 and 
give each the same sign as the t from which it is derived 

Then 

2 q = l t 2 1 * r 



136 A THEORY OF PROBABLE INFERENCE. 



TaUeofSt = 



p 



dt. 



t 


0; 


0.0 


0.000 


0.1 


0.112 


0.2 


0.223 


0.3 


0.329 


0.4 


0.428 


0.5 


0.520 


0.6 


0.604 


0.7 


0.678 


0.8 


0.742 


0.9 


0.797 


1.0 


0.843 



t 


et 


1.0 


0.843 


1.1 


0.880 


1.2 


0.910 


1.3 


0.934 


1.4 


0.952 


1.5 


0.966 


1.6 


0.976 


1.7 


0.984 


1.8 


0.989 


1.9 


0.993 


2.0 


0.995 



t 


0* 


2.0 


0.99532 


2.1 


0.99702 


2.2 


0.99814 


2.3 


0.9988"6 


2.4 


0.99931 


2.5 


0.99959 


2.6 


0.99976 


2.7 


0.99987 


2.8 


0.99992 


2.9 


0.99996 


3.0 


0.99998 



t 





4 


0.999999989 


5 


0.9999999999984 


6 


0.999999999999999982 


7 


0.999999999999999999999958 



In rough calculations we may take t equal to t for t 
less than 0.7, and as equal to unity for any value above 
t = 1.4. 

The principle of statistical deduction is that these two 
proportions, namely, that of the P s among the if s, 
and that of the P s among the s, are probably and 
approximately equal. If, then, this principle justifies our 
inferring the value of the second proportion from the 
known value of the first, it equally justifies our inferring 
the value of the first from that of the second, if the first 



A THEORY OF PROBABLE INFERENCE. 137 

is unknown but the second has been observed. We 
thus obtain the following form of inference : 

FORM V. 
Induction. 

S 1 , S fl , S ", etc., form a numerous set taken at random 
from among the M s ; 

S 1 , S", S ", etc., are found to be the proportion p of 
them P s : 

Hence, probably and approximately the same proportion, p, 
of the M a are P s. 

The following are examples. From a bag of coffee a 
handful is taken out, and found to have nine tenths of 
the beans perfect ; whence it is inferred that about nine- 
tenths of all the beans in the bag are probably perfect. 
The United States Census of 1870 shows that of native 
white children under one year old, there were 478,774 
males to 463,320 females ; while of colored children of 
the same age there were 75,985 males to 76,637 females. 
We infer that generally there is a larger proportion of 
female births among negroes than among whites. 

When the ratio p is unity or zero, the inference is an 
ordinary induction ; and 1 ask leave to extend the term 
induction to all such inference, whatever be the value of 
p. It is, in fact, inferring from a sample to the whole 
lot sampled. These two forms of inference, statistical 
deduction and induction, plainly depend upon the same 
principle of equality of ratios, so that their validity is the 
same. Yet the nature of the probability in the two cases 
is very different. In the statistical deduction, we know 
that among the whole body of M s the proportion of P s 
is p ; we say, then, that the S 9 s being random drawings 



138 A THEORY OF PROBABLE INFERENCE. 

of Jf sare probably P s in about the same proportion, 
and though this may happen not to be so, yet at any 
rate, on continuing the drawing sufficiently, our pre 
diction of the ratio will be vindicated at last. On the 
other hand, in induction we say that the proportion p of 
the sample being P s, probably there is about the same 
proportion in the whole lot ; or at least, if this happens 
not to be so, then on continuing the drawings the in 
ference will be, not vindicated as in the other case, but 
modified so as to become true. The deduction, then, 
is probable in this sense, that though its conclusion may 
in a particular case be falsified, yet similar conclusions 
(with the same ratio p) would generally prove approxi 
mately true; while the induction is probable in this 
sense, that though it may happen to give a false con 
clusion, yet in most cases in which the same precept of 
inference was followed, a different and approximately 
true inference (with the right value of p) would be 
drawn. 



IY. 

Before going any further with the study of Form V., 
I wish to join to it another extremely analogous form. 

We often speak of one thing being very much like 
another, and thus apply a vague quantity to resemblance. 
Even if qualities are not subject to exact numeration, 
we may conceive them to be approximately measurable. 
We may then measure resemblance by a scale of num 
bers from zero up to unity. To say that S has a 
1-likeness to a P will mean that it has every character 
of a P, and consequently is a P. To say that it has a 
0-likeness will imply total dissimilarity. We shall then 
be able to reason as follows : 



A THEORY OF PROBABLE INFERENCE. 139 

FORM II. (bis). 
Simple probable deduction in depth. 

Every M has the simple mark P ; 
The >S"s have an r-likeness to the M s : 
Hence, the probability is r that every S is P. 

It would be difficult, perhaps impossible, to adduce an 
example of such kind of inference, for the reason that 
simple marks are not known to us. We may, however, 
illustrate the complex probable deduction in depth (the 
general form of which it is not worth while to set down) 
as follows : I forget whether, in the ritualistic churches, 
a bell is tinkled at the elevation of the Host or not. 
Knowing, however, that the services resemble somewhat 
decidedly those of the Roman Mass, I think that it is not 
unlikely that the bell is used in the ritualistic, as in the 
Roman, churches. 

We shall also have the following : 

FORM IV. (bis). 
Statistical deduction in depth. 

Every M has, for example, the numerous marks P 7 , P rr , 
P", etc. 

S has an r-likeness to the M s : 

Hence, probably and approximately, S has the proportion r 
of the marks P , P", P ", etc. 

For example, we know that the French and Italians 
are a good deal alike in their ideas, characters, tempera 
ments, genius, customs, institutions, etc., while they also 
differ very markedly in all these respects. Suppose, then, 
that I know a boy who is going to make a short trip 
through France and Italy ; I can safely predict that 
among the really numerous though relatively few res- 



140 A THEORY OF PROBABLE INFERENCE. 

pects in which he will be able to compare the two people, 
about the same degree of resemblance will be found. 

Both these modes of inference are clearly deductive. 
When r = 1, they reduce to Barbara. 1 

Corresponding to induction, we have the following 
mode of inference: 

FORM V. (bis). 
Hypothesis. 

M has, for example, the numerous marks P , P", P /;/ , etc. 
S has the proportion r of the marks P , P", P" ! , etc. : 
Hence, probably and approximately, has an r-likeness to M. 

Thus, we know, that the ancient Mound-builders of 
North America present, in all those respects in which we 
have been able to make the comparison, a limited degree 
of resemblance with the Pueblo Indians. The inference 
is, then, that in all respects there is about the same de 
gree of resemblance between these races. 

If I am permitted the extended sense which I have 
given to the word " induction," this argument is simply 
an induction respecting qualities instead of respecting 

1 When r = 0, the last form becomes 

M has all the marks P ; 

S has no mark of M : 

Hence, S has none of the marks P. 

"When the universe of marks is unlimited (see a note appended to this 
paper for an explanation of this expression), the only way in which two 
terms can fail to have a common mark is by their together filling the uni 
verse of things ; and consequently this form then becomes, 

3/isP; 

Every non- is M: 

Hence, every non-S is P. 

This is one of De Moi-gan s syllogisms. 

In putting r = in Form II. (bis) it must be noted that, since P is 
simple in depth, to say that S is not P is to say that it has no mark of P. 



A THEORY OF PROBABLE INFERENCE. 141 

things. In point of fact P , P", P" , etc. constitute a 
random sample of the characters of M, and the ratio r 
of them being found to belong to $ the same ratio of all 
the characters of M are concluded to belong to S. This 
kind of argument, however, as it actually occurs, differs 
very much from induction, owing to the impossibility 
of simply counting qualities as individual things are 
counted. Characters have to be weighed rather than 
counted. Thus, antimony is bluish-gray : that is a char 
acter. Bismuth is a sort of rose-gray; it is decidedly 
different from antimony in color, and yet not so very 
different as gold, silver, copper, and tin are. 

I call this induction of characters hypothetic inference, 
or, briefly, hypothesis. This is perhaps not a very happy 
designation, yet it is difficult to find a better. The term 
"hypothesis" has many well established and distinct 
meanings. Among these is that of a proposition believed 
in because its consequences agree with experience. This 
is the sense in which Newton used the word when he 
said, Hypotheses non jingo. He meant that he was merely 
giving a general formula for the motions of the heavenly 
bodies, but was not undertaking to mount to the causes 
of the acceleration they exhibit. The inferences of 
Kepler, on the other hand, were hypotheses in this sense; 
for he traced out the miscellaneous consequences of the 
supposition that Mars moved in an ellipse, with the sun 
at the focus, and showed that both the longitudes and the 
latitudes resulting from this theory were such as agreed 
with observation. These two components of the motion 
were observed ; the third, that of approach to or regression 
from the earth, was supposed. Now, if in Form V. (bis) 
we put r = 1, the inference is the drawing of a hypothesis 
in this sense. I take the liberty of extending the use of 
the word by permitting r to have any value from zero to 



142 A THEORY OF PKOBABLE INFERENCE. 

unity. The term is certainly not all that could be de 
sired ; for the word hypothesis, as ordinarily used, carries 
with it a suggestion of uncertainty, and of something to 
be superseded, which does not belong at all to my use of 
it. But we must use existing language as best we may, 
balancing the reasons for and against any mode of ex 
pression, for none is perfect ; at least the term is not 
so utterly misleading as " analogy " would be, and with 
proper explanation it will, I hope, be understood. 

y. 

The following examples will illustrate the distinction 
between statistical deduction, induction, and hypothesis. 
If I wished to order a font of type expressly for the 
printing of this book, knowing, as I do, that in all Eng 
lish writing the letter e occurs oftener than any other 
letter, I should want more e s in my font than other 
letters. For what is true of all other English writing is 
no doubt true of these papers. This is a statistical de 
duction. But then the words used in logical writings are 
rather peculiar, and a good deal of use is made of single 
letters. I might, then, count the number of occurrences 
of the different letters upon a dozen or so pages of the 
manuscript, and thence conclude the relative amounts of 
the different kinds of type required in the font. That 
would be inductive inference. If now I were to order 
the font, and if, after some days, I were to receive a box 
containing a large number of little paper parcels of very 
different sizes, I should naturally infer that this was the 
font of types I had ordered ; and this would be hypothetic 
inference. Again, if a dispatch in cipher is captured, and 
it is found to be written with twenty-six characters, one 
of which occurs much more frequently than any of the 



A THEORY OF PROBABLE INFERENCE. 143 

others, we are at once led to suppose that each charac 
ter represents a letter, and that the one occurring so fre 
quently stands fer e. This is also hypothetic inference. 

We are thus led to divide all probable reasoning into 
deductive and ampliative, and further to divide ampliative 
reasoning into induction and hypothesis. In deductive 
reasoning, though the predicted ratio may be wrong in a 
limited number of drawings, yet it will be approximately 
verified in a larger number. In ampliative reasoning the 
ratio may be wrong, because the inference is based on but 
a limited number of instances ; but on enlarging the 
sample the ratio will be changed till it becomes approxi 
mately correct. In induction, the instances drawn at 
random are numerable things ; in hypothesis they are 
characters, which are not capable of strict enumeration, 
but have to be otherwise estimated. 

This classification of probable inference is connected 
with a preference for the copula of inclusion over those 
used by Miss Ladd and by Mr. Mitchell. 1 De Morgan 
established eight forms of simple propositions ; and from 
a purely formal point of view no one of these has a right 
to be considered as more fundamental than any other. 
But formal logic must not be too purely formal ; it must 
represent a fact of psychology, or else it is in danger of 
degenerating into a mathematical recreation. The cate 
gorical proposition, "every man is mortal," is but a modifi 
cation of the hypothetical proposition, " if humanity, then 
mortality ;" and since the very first conception from which 
logic springs is that one proposition follows from another, 
I hold that "if A, then B" should be taken as the typical 
form of judgment. Time flows ; and, in time, from one 
state of belief (represented by the premises of an argu- 

1 I do not here speak of Mr. Jeyons, because my objection to the copula 
of identity is of a somewhat different kind. 



144 A THEOKY OF PROBABLE INFERENCE. 

ment) another (represented by its conclusion) is de 
veloped. Logic arises from this circumstance, without 
which we could not learn anything nor correct any 
opinion. To say that an inference is correct is to say 
that if the premises are true the conclusion is also true ; 
or that every possible state of things in which the prem 
ises should be true would be included among the possible 
states of things in which the conclusion would be true. 
We are thus led to the copula of inclusion. But the 
main characteristic of the relation of inclusion is that it 
is transitive, that is, that what is included in some 
thing included in anything is itself included in that 
thing ; or, that if A is B and B is (7, then A is 0. We 
thus get Barbara as the primitive type of inference. 
Now in Barbara we have a Rule, a Case under the Rule, 
and the inference of the Result of that rule in that case. 
For example : 

Rule. All men are mortal ; 
Case. Enoch was a man. 
Result. Enoch was mortal. 

The cognition of a rule is not necessarily conscious, 
but is of the nature of a habit, acquired or congenital. 
The cognition of a case is of the general nature of a 
sensation; that is to say, it is something which comes 
up into present consciousness. The cognition of a result 
is of the nature of a decision to act in a particular way 
on a given occasion. 1 In point of fact, a syllogism, in 
Barbara virtually takes place when we irritate the foot 
of a decapitated frog. The connection between the af 
ferent and efferent nerve, whatever it may be, constitutes 
a nervous habit, a rule of action, which is the physio- 

1 See my paper on " How to make our ideas clear." Popular Science 
Monthly, January, 1878. 



A THEORY OF PROBABLE INFERENCE. 145 

logical analogue of the major premise. The disturbance 
of the ganglionic equilibrium, owing to the irritation, is 
the physiological form of that which, psychologically con 
sidered, is a sensation ; and, logically considered, is the 
occurrence of a case. The explosion through the efferent 
nerve is the physiological form of that which psychologi 
cally is a volition, and logically the inference of a result. 
When we pass from the lowest to the highest forms of 
inervation, the physiological equivalents escape our ob 
servation ; but, psychologically, we still have, first, habit, 
which in its highest form is understanding, and which 
corresponds to the major premise of Barbara; we have, 
second, feeling, or present consciousness, corresponding 
to the minor premise of Barbara; and we have, third, 
volition, corresponding to the conclusion of the same 
mode of syllogism. Although these analogies, like all 
very broad generalizations, may seem very fanciful at 
first sight, yet the more the reader reflects upon them 
the more profoundly true I am confident they will appear. 
They give a significance to the ancient system of formal 
logic which no other can at all share. 

Deduction proceeds from Rule and Case to Result ; it 
is the formula of Volition. Induction proceeds from Case 
and Result to Rule ; it is the formula of the formation of 
a habit or general conception, a process which, psycho 
logically as well as logically, depends on the repetition of 
instances or sensations. Hypothesis proceeds from Rule 
and Result to Case ; it is the formula of the acquirement 
of secondary sensation, a process by which a confused 
concatenation of predicates is brought into order under 
a synthetizing predicate. 

We usually conceive Nature to be perpetually making 
deductions in Barbara. This is our natural and anthro 
pomorphic metaphysics. We conceive that there are 



146 A THEOKY OF PROBABLE INFERENCE. 

Laws of Nature, which are her Rules or major premises. 
We conceive that Cases arise under these laws ; these 
cases consist in the predication, or occurrence, of causes, 
which are the middle terms of the syllogisms. And, 
finally, we conceive that the occurrence of these causes, 
by virtue of the laws of Nature, result in effects which 
are the conclusions of the syllogisms. Conceiving of 
nature in this way, we naturally conceive of science as 
having three tasks, (1) the discovery of Laws, which 
is accomplished by induction ; (2) the discovery of Causes, 
which is accomplished by hypothetic inference ; and (3) 
the prediction of Effects, which is accomplished by de 
duction. It appears to me to be highly useful to select 
a system of logic which shall preserve all these natural 
conceptions. 

It may be added that, generally speaking, the conclu 
sions of Hypothetic Inference cannot be arrived at in 
ductively, because their truth is not susceptible of direct 
observation in single cases. Nor can the conclusions of 
Inductions, on account of their generality, be reached by 
hypothetic inference. For instance, any historical fact, 
as that Napoleon Bonaparte once lived, is a hypothesis ; 
we believe the fact, because its effects I mean current 
tradition, the histories, the monuments, etc. are ob 
served. But no mere generalization of observed facts 
could ever teach us that Napoleon lived. So we induc 
tively infer that every particle of matter gravitates toward 
every other. Hypothesis might lead to this result for 
any given pair of particles, but it never could show that 
the law was universal. 

VI. 

We now come to the consideration of the Rules which 
have to be followed in order to make valid and strong 



A THEOKY OF PKOBABLE INFERENCE. 147 

Inductions and Hypotheses. These rules can all be re 
duced to a single one ; namely, that the statistical deduc 
tion of which the Induction or Hypothesis is the inversion, 
must be valid and strong. 

We have seen that Inductions and Hypotheses are in 
ferences from the conclusion and one premise of a sta 
tistical syllogism to the other premise. In the case of 
hypothesis, this syllogism is called the explanation. Thus 
in one of the examples used above, we suppose the cryp 
tograph to be an English cipher, because, as we say, this 
explains the observed phenomena that there are about 
two dozen characters, that one occurs more frequently 
than the rest, especially at the ends of words, etc. The 
explanation is, 

Simple English ciphers have certain peculiarities ; 

This is a simple English cipher : 

Hence, this necessarily has these peculiarities. 

This explanation is present to the mind of the reasoner, 
too ; so much so, that we commonly say that the hypo 
thesis is adopted for the sake of the explanation. Of 
induction we do not, in ordinary language, say that it 
explains phenomena; still, the statistical deduction, of 
which it is the inversion, plays, in a general way, the 
same part as the explanation in hypothesis. From a 
barrel of apples, that I am thinking of buying, I draw 
out three or four as a sample. If I find the sample some 
what decayed, I ask myself, in ordinary language, not 
Why is this ? " but How is this ? And I answer 
that it probably comes from nearly all the apples in the 
barrel being in bad condition. The distinction between 
the Why" of hypothesis and the " How" of induction 
is not very great ; both ask for a statistical syllogism, of 
which the observed fact shall be the conclusion, the 



148 A THEOEY OF PROBABLE INFERENCE. 

known conditions of the observation one premise, and 
the inductive or hypothetic inference the other. This 
statistical syllogism may be conveniently termed the ex 
planatory syllogism. 

In order that an induction or hypothesis should have 
any validity at all, it is requisite that the explanatory 
syllogism should be a valid statistical deduction. Its 
conclusion must not merely follow from the premises, 
but follow from them upon the principle of probability. 
The inversion of ordinary syllogism does not give rise 
to an induction or hypothesis. The statistical syllogism 
of Form 1Y. is invertlble, because it proceeds upon the 
principle of an approximate equality between the ratio 
of P s in the whole class and the ratio in a well-drawn 
sample, and because equality is a convertible relation. 
But ordinary syllogism is based upon the property of the 
relation of containing and contained, and that is not a 
convertible relation. There is, however, a way in which 
ordinary syllogism may be inverted ; namely, the con 
clusion and either of the premises may be interchanged 
by negativing each of them. This is the way in which 
the indirect, or apagogical, 1 figures of syllogism are de 
rived from the first, and in which the modus tollens is 
derived from the modus ponens. The following schemes 

show this : 

First Figure. 

Rule. AllJfisP; 
Case. S is M : 
Result. S is P. 



Second Figure. 
Rule. AllJfisP; 

Denial of Result. S is not P : 
Denial of Case. S is not M. 



Third Figure. 

Denial of Result. S is not P ; 
Case. /Sis M: 

Denial of Rule. Some M is 



not P. 

1 From apagoge, Aristotle s name for the rcductio ad alsurdum. 



A THEORY OF PROBABLE INFERENCE. 149 

Modus Ponens. 

Rule. If A is true, C is true ; 
Case. In a certain case A is true : 
Result. . . In that case C is true. 



Modus Tollens. 

Rule. If A is true, C is 

true; 
Denial of Result. In a certain 

case C is not true : 
Denial of Case. . . In that 

case A is not true. 



Modus Innominatus. 

Case. In a certain case A is 

true; 
Denial of Result. In that case 

C is not true : 
Denial of Rule. . . If A is true, 

C is not necessarily true. 



Now suppose we ask ourselves what would be the re 
sult of thus apagogically inverting a statistical deduction. 
Let us take, for example, Form IV : 

The $ s are a numerous random sample of the M s ; 

The proportion r of the M s are P s : 

Hence, probably about the proportion r of the S s are P s. 

The ratio r, as we have already noticed, is not neces 
sarily perfectly definite ; it may be only known to have 
a certain maximum or minimum ; in fact, it may have 
any kind of indeterminacy. Of all possible values be 
tween and 1, it admits of some and excludes others. 
The logical negative of the ratio r is, therefore, itself a 
ratio, which we may name p ; it admits of every value 
which r excludes, and excludes every value of which r 
admits. Transposing, then, the major premise and con 
clusion of our statistical deduction, and at the same time 
denying both, we obtain the following inverted form : 



150 A THEORY OF PROBABLE INFERENCE. 

The S s are a numerous random sample of the M a ; 

The proportion p of the S s are .P s : 

Hence, probably about the proportion p of the M a are P s. 1 

But this coincides with the formula of Induction. 
Again, let us apagogically invert the statistical deduction 
of Form IV. (fo s). This form is, 

Every M has, for example, the numerous marks P ; , P", 
P" , etc. 

S has an r-likeness to the M a : 

Hence, probably and approximately, S has the proportion 
r of the marks P f , P", P" 1 , etc. 

Transposing the minor premise and conclusion, at the 
same time denying both, we get the inverted form, 

Every M has, for example, the numerous marks P f , P rf , 
P" , etc. 

S has the proportion p of the marks P , P", P f!f , etc. : 
Hence, probably and approximately, S has a p-likeness to 

the class of M ? s. 

This coincides with the formula of Hypothesis. Thus 
we see that Induction and Hypothesis are nothing but 
the apagogical inversions of statistical deductions. Ac 
cordingly, when r is taken as 1, so that p is "less than 1," 
or when r is taken as 0, so that p is " more than 0," the 
induction degenerates into a syllogism of the third figure 
and the hypothesis into a syllogism of the second figure. 

1 The conclusion of the statistical deduction is here regarded as being 
"the proportion r of the S s are P s," and the words "probably about" 
as indicating the modality with which this conclusion is drawn and held 
for true. It would be equally true to consider the "probably about" as 
forming part of the contents of the conclusion ; only from that point of 
view the inference ceases to be probable, and becomes rigidly necessary, 
and its apagogical inversion is also a necessary inference presenting no 
particular interest. 



A THEORY OF PEOBABLE INFERENCE. 151 

In these special cases, there is no very essential difference 
between the mode of reasoning in the direct and in the 
apagogical form. But, in general, while the probability 
of the two forms is precisely the same, in this sense, 
that for any fixed proportion of _P s among the M a 
(or of marks of jS 9 s among the marks of the M s) the 
probability of any given error in the concluded value is 
precisely the same in the indirect as it is in the direct 
form, yet there is this striking difference, that a multi 
plication of instances will in the one case confirm, and 
in the other modify, the concluded value of the ratio. 

We are thus led to another form for our rule of validity 
of ampliative inference ; namely, instead of saying that 
the explanatory syllogism must be a good probable de 
duction, we may say that the syllogism of which the 
induction or hypothesis is the apagogical modification 
(in the traditional language of logic, the reduction) must 
be valid. 

Probable inferences, though valid, may still differ in 
their strength. A probable deduction has ,a greater or 
less probable error in the concluded ratio. When r is a 
definite number the probable error is also definite ; but 
as a general rule we can only assign maximum and mini 
mum values of the probable error. The probable error 
is, in fact, 



0.477 V^-^ 
n 

where n is the number of independent instances. The 
same formula gives the probable error of an induction or 
hypothesis ; only that in these cases, r being wholly inde 
terminate, the minimum value is zero, and the maximum 
is obtained by putting r = J. 



152 A THEORY OF PROBABLE INFERENCE. 

VII. 

Although the rule given above really contains all the 
conditions to which Inductions and Hypotheses need to 
conform, yet inasmuch as there are many delicate ques 
tions in regard to the application of it, and particularly 
since it is of that nature that a violation of it, if not 
too gross, may not absolutely destroy the virtue of the 
reasoning, a somewhat detailed study of its requirements 
in regard to each of the premises of the argument is still 
needed. 

The first premise of a scientific inference is that certain 
things (in the case of induction) or certain characters 
(in the case of hypothesis) constitute a fairly chosen 
sample of the class of things or the run of characters 
from which they have been drawn. 

The rule requires that the sample should be drawn at 
random and independently from the whole lot sampled. 
That is to say, the sample must be taken according to a 
precept or method which, being applied over and over 
again indefinitely, would in the long run result in the 
drawing of any one set of instances as often as any other 
set of the same number. 

The needfulness of this rule is obvious ; the difficulty 
is to know how we are to carry it out. The usual method 
is mentally to run over the lot of objects or characters to 
be sampled, abstracting our attention from their peculi 
arities, and arresting ourselves at this one or that one 
from motives wholly unconnected with those peculiarities. 
But this abstention from a further determination of our 
choice often demands an effort of the will that is beyond 
our strength ; and in that case a mechanical contrivance 
may be called to our aid. We may, for example, number 
all the objects of the lot, and then draw numbers by 



A THEORY OF PROBABLE INFERENCE. 153 

means of a roulette, or other such instrument. We may 
even go so far as to say that this method is the type of 
all random drawing ; for when we abstract our attention 
from the peculiarities of objects, the psychologists tell us 
that what we do is to substitute for the images of sense 
certain mental signs, and when we proceed to a random 
and arbitrary choice among these abstract objects we are 
governed by fortuitous determinations of the nervous sys 
tem, which in this case serves the purpose of a roulette. 

The drawing of objects at random is an act in which 
honesty is called for ; and it is often hard enough to be 
sure that we have dealt honestly with ourselves in the 
matter, and still more hard to be satisfied of the honesty 
of another. Accordingly, one method of sampling has 
come to be preferred in argumentation ; namely, to take 
of the class to be sampled all the objects of which we 
have a sufficient knowledge. Sampling is, however, a 
real art, well deserving an extended study by itself : to 
enlarge upon it here would lead us aside from our main 
purpose. 

Let us rather ask what will be the effect upon inductive 
inference of an imperfection in the strictly random char 
acter of the sampling. Suppose that, instead of using 
such a precept of selection that any one M would in the 
long run be chosen as often as any other, we used a 
precept which would give a preference to a certain half 
of the J/ s, so that they would be drawn twice as often 
as the rest. If we were to draw a numerous sample by 
such a precept, and if we were to find that the proportion 
p of the sample consisted of JP s, the inference that we 
should be regularly entitled to make would be, that among 
all the M a, counting the preferred half for two each, the 
proportion p would be P s. But this regular inductive 
inference being granted, from it we could deduce by 



154 A THEORY OF PROBABLE INFERENCE. 

arithmetic the further conclusion that, counting the M s 
for one each, the proportion of P s among them must 
(p being over f ) lie between | p + \ and f p J. Hence, 
if more than two thirds of the instances drawn by the use 
of the false precept were found to be P s, we should be 
entitled to conclude that more than half of all the M s 
were P s. Thus, without allowing ourselves to be led 
away into a mathematical discussion, we can easily see 
that, in general, an imperfection of that kind in the 
random character of the sampling will only weaken the 
inductive conclusion, and render the concluded ratio less 
determinate, but will not necessarily destroy the force 
of the argument completely. In particular, when p ap 
proximates towards 1 or 0, the effect of the imperfect 
sampling will be but slight. 

Nor must we lose sight of the constant tendency of the 
inductive process to correct itself. This is of its essence. 
This is the marvel of it. The probability of its conclusion 
only consists in the fact that if the true value of the ratio 
sought has not been reached, an extension of the induc 
tive process will lead to a closer approximation. Thus, 
even though doubts may be entertained whether one se 
lection of instances is a random one, yet a different se 
lection, made by a different method, will be likely to vary 
from the normal in a different way, and if the ratios 
derived from such different selections are nearly equal, 
they may be presumed to be near the truth. This con 
sideration makes it extremely advantageous in all ampli- 
ative reasoning to fortify one method of investigation by 
another. 1 Still we must not allow ourselves to trust so 

1 This I conceive to be all the truth there is in the doctrine of Bacon 
and Mill regarding different Methods of Experimental Inquiry. The main 
proposition of Bacon and Mill s doctrine is, that in order to prove that all 
M a are P s, we should not only take random instances of the M s and 



A THEORY OF PROBABLE INFERENCE. 155 

much to this virtue of induction as to relax our efforts 
towards making our drawings of instances as random 
and independent as we can. For if we infer a ratio from 
a number of different inductions, the magnitude of its 
probable error will depend very much more on the worst 
than on the best inductions used. 

We have, thus far, supposed that although the selection 
of instances is not exactly regular, yet the precept fol 
lowed is such that every unit of the lot would eventually 
get drawn. But very often it is impracticable so to draw 
our instances, for the reason that a part of the lot to be 
sampled is absolutely inaccessible to our powers of obser 
vation. If we want to know whether it will be profit 
able to open a mine, we sample the ore ; but in advance 
of our mining operations, we can obtain only what ore 
lies near the surface. Then, simple induction becomes 
worthless, and another method must be resorted to. Sup 
pose we wish to make an induction regarding a series 
of events extending from the distant past to the distant 
future ; only those events of the series which occur within 
the period of time over which available history extends 
can be taken as instances. Within this period we may 
find that the events of the class in question present some 
uniform character ; yet how do we know but this uni 
formity was suddenly established a little while before the 
history commenced, or will suddenly break up a little 
while after it terminates ? Now, whether the uniformity 

examine them to see that they are Ps, but we should also take instances 
of not-P s and examine them to see that they are not-J/ s. This is an 
excellent way of fortifying one induction by another, when it is applicable; 
but it is entirely inapplicable when r has any other value than 1 or 0. 
For, in general, there is no connection between the proportion of M s that 
are Ps and the proportion of non-P s that are non-l/ s. A very small 
proportion of calves may be monstrosities, and yet a very large proportion 
of monstrosities may be calves. 



156 A THEORY OF PROBABLE INFERENCE. 

observed consists (1) in a mere resemblance between all 
the phenomena, or (2) in their consisting of a disorderly 
mixture of two kinds in a certain constant proportion, or 
(3) in the character of the events being a mathematical 
function of the time of occurrence, in any of these cases 
we can make use of an apagoge from the following proba 
ble deduction : 

Within the period of time M, a certain event P occurs ; 
S is a period of time taken at random from M, and more 
than half as long : 

Hence, probably the event P will occur within the time S. 

Inverting this deduction, we have the following ampli- 
ative inference : 

S is a period of time taken at random from M 9 and more 

than half as long ; 

The event P does not happen in the time S : 

Hence, probably the event P does not happen in the 

period M. 

The probability of the conclusion consists in this, that 
we here follow a precept of inference, which, if it is very 
often applied, will more than half the time lead us right. 
Analogous reasoning would obviously apply to any por 
tion of an unidimensional continuum, which might be 
similar to periods of time. This is a sort of logic which 
is often applied by physicists in what is called extrapola 
tion of an empirical law. As compared with a typical 
induction, it is obviously an excessively weak kind of in 
ference. Although indispensable in almost every branch 
of science, it can lead to no solid conclusions in regard to 
what is remote from the field of direct perception, unless 
it be bolstered up in certain ways to which we shall have 
occasion to refer further on. 



A THEORY OF PROBABLE INFERENCE. 157 

Let us now consider another class of difficulties in 
regard to the rule that the samples must be drawn at 
random and independently. In the first place, what if 
the lot to be sampled be infinite in number ? In what 
sense could a random sample be taken from a lot like 
that ? A random sample is one taken according to a 
method that would, in the long run, draw any one object 
as often as any other. In what sense can such drawing 
be made from an infinite class ? The answer is not far 
to seek. Conceive a cardboard disk revolving in its own 
plane about its centre, and pretty accurately balanced, 
so that when put into rotation it shall be about 1 as likely 
to come to rest in any one position as in any other ; and 
let a fixed pointer indicate a position on the disk: the 
number of points on the circumference is infinite, and on 
rotating the disk repeatedly the pointer enables us to 
make a selection from this infinite number. Tbis means 
merely that although the points are innumerable, yet 
there is a certain order among them that enables us to 
run them through and pick from them as from a very 
numerous collection. In such a case, and in no other, 
can an infinite lot be sampled. But it would be equally 
true to say that a finite lot can be sampled only on 
condition that it can be regarded as equivalent to an 
infinite lot. For the random sampling of a finite class 
supposes the possibility of drawing out an object, throw 
ing it back, and continuing this process indefinitely ; so 
that what is really sampled is not the finite collection of 
things, but the unlimited number of possible drawings. 

But though there is thus no insuperable difficulty in 
sampling an infinite lot, yet it must be remembered that 
the conclusion of inductive reasoning only consists in the 

1 I say about, because the doctrine of probability only deals with ap 
proximate evaluations. 



158 A THEORY OF PROBABLE INFERENCE. 

approximate evaluation of a ratio, so that it never can 
authorize us to conclude that in an infinite lot sampled 
there exists no single exception to a rule. Although all 
the planets are found to gravitate toward one another, 
this affords not the slightest direct reason for denying 
that among the innumerable orbs of heaven there may 
be some \vhich exert no such force. Although at no 
point of space where we have yet been have we found 
any possibility of motion in a fourth dimension, yet this 
does not tend to show (by simple induction, at least) 
that space has absolutely but three dimensions. Although 
all the bodies we have had the opportunity of examining 
appear to obey the law of inertia, this does not prove 
that atoms and atomicules are subject to the same law. 
Such conclusions must be reached, if at all, in some 
other way than by simple induction. This latter may 
show that it is unlikely that, in my lifetime or yours, 
things so extraordinary should be found, but do not war 
rant extending the prediction into the indefinite future. 
And experience shows it is not safe to predict that such 
and such a fact will never be met with. 

If the different instances of the lot sampled are to 
be drawn independently, as the rule requires, then the 
fact that an instance has been drawn once must not 
prevent its being drawn again. It is true that if the 
objects remaining unchosen are very much more numer 
ous than those selected, it makes practically no difference 
whether they have a chance of being drawn again or not, 
since that chance is in any case very small. Proba 
bility is wholly an affair of approximate, not at all of 
exact, measurement ; so that when the class sampled is 
very large, there is no need of considering whether ob 
jects can be drawn more than once or not. But in what 
is known as " reasoning from analogy," the class sam- 



A THEORY OF PROBABLE INFERENCE. 159 

pled is small, and no instance is taken twice. For ex 
ample : we know that of the major planets the Earth, 
Mars, Jupiter, and Saturn revolve on their axes, and 
we conclude that the remaining four, Mercury, Venus, 
Uranus, and Neptune, probably do the like. This is 
essentially different from an inference from what has 
been found in drawings made hitherto, to what will be 
found in indefinitely numerous drawings to be made 
hereafter. Our premises here are that the Earth, Mars, 
Jupiter, and Saturn are a random sample of a natural 
class of major planets, a class which, though (so far 
as we know) it is very small, yet may be very extensive, 
comprising whatever there may be that revolves in a 
circular orbit around a great sun, is nearly spherical, 
shines with reflected light, is very large, etc. Now the 
examples of major planets that we can examine all ro 
tate on their axes ; whence we suppose that Mercury, 
Venus, Uranus, and Neptune, since they possess, so far 
as we know, all the properties common to the natural 
class to which the Earth, Mars, Jupiter, and Saturn be 
long, possess this property likewise. The points to be 
observed are, first, that any small class of things may be 
regarded as a mere sample of an actual or possible large 
class having the same properties and subject to the same 
conditions; second, that while we do not know what all 
these properties and conditions are, we do know some of 
them, which some may be considered as a random sam 
ple of all ; third, that a random selection without re 
placement from a small class may be regarded as a true 
random selection from that infinite class of which the 
finite class is a random selection. The formula of the 
analogical inference presents, therefore, three premises, 
thus : 



160 A THEOEY OF PROBABLE INFERENCE. 

/S 7 , S", S" f are a random sample of some undefined class X } 
of whose characters P f , P", P" 1 are samples. 
Q is P, P", P ". 
S , S", S">, are 7? s. 
Hence, Q is an R. 

"We have evidently here an induction and an hypothe 
sis followed by a deduction ; thus, 



Every X is, for example, P , 
P", P 7 , etc. 

Q is found to be P , P", 
P>", etc. 

Hence, hypothetically, Q is 



*S f/ , ", /S^ , etc., are samples 



of the 



S , S", S>, etc., are found 
to be It s. 

Hence, inductively, every X 



is an R. 
Hence, deductively, Q is an R.* 

An argument from analogy may be strengthened by 
the addition of instance after instance to the premises, 
until it loses its ampliative character by the exhaustion 
of the class and becomes a mere deduction of that kind 
called complete induction, in which, however, some shadow 

* That this is really a correct analysis of the reasoning can be shown by 
the theory of probabilities. For the expression 

(P + g) ! (TT + P) (P + ^)! (? + P)! 

p \ q ! 7T ! p ! (p -j- TT -{- g -f- p) ! 

expresses at once the probability of two events ; namely, it expresses 
first the probability that of p -f <7 objects drawn without replacement 
from a lot consisting of p -f TT objects having the character E together 
with q -f- p not having this character, the number of those drawn having 
this character will bejo; and second, the same expression denotes the 
probability that if among p -f- TT -f- q -f- p objects drawn at random from 
an infmita class (containing no matter what proportion of It s to nori-72 s), 
it happens that p -f- TT have the character 72, then among any ^ -f- g of 
them, designated at random, p will have the same character. Thus we 
see that the chances in reference to drawing without replacement from a 
finite class are precisely the same as those in reference to a class which 
has been drawn at random from an infinite class. 



A THEOKY OF PROBABLE INFEKENCE. 161 

of the inductive character remains, as this name im 
plies. 

VIII. 

Take any human being, at random, say Queen Eliz 
abeth. Now a little more than half of all the human 
beings who have ever existed have been males ; but it 
does not follow that it is a little more likely than not 
that Queen Elizabeth was a male, since we know she was 
a woman. Nor, if we had selected Julius Caesar, would 
it be only a little more likely than not that he was a 
male. It is true that if we were to go on drawing at 
random an indefinite number of instances of human be 
ings, a slight excess over one-half would be males. But 
that which constitutes the probability of an inference is 
the proportion of true conclusions among all those which 
could be derived from the same precept. Now a precept 
of inference, being a rule which the mind is to follow, 
changes its character and becomes different when the 
case presented to the mind is essentially different. When, 
knowing that the proportion r of all M* s are P s, I draw 
an instance, S, of an M, without any other knowledge of 
whether it is a P or not, and infer with probability, r, 
that it is P, the case presented to my mind is very 
different from what it is if I have such other knowledge. 
In short, I cannot make a valid probable inference with 
out taking into account whatever knowledge I have (or, 
at least, whatever occurs to my mind) that bears upon 
the question. 

The same principle may be applied to the statistical 
deduction of Form IV. If the major premise, that the 
proportion r of the 3/ s are P s, be laid down first, 
before the instances of Ms are drawn, we really draw our 
inference concerning those instances (that the proper- 



162 A THEORY OF PROBABLE INFERENCE. 

tion r of them will be P s) in advance of the drawing, 
and therefore before we know whether they are P s or 
not. But if we draw the instances of the M B first, and 
after the examination of them decide what we will select 
for the predicate of our major premise, the inference 
will generally be completely fallacious. In short, we 
have the rule that the major term P must be decided 
upon in advance of the examination of the sample ; and 
in like manner in Form IV. (bis) the minor term S must 
be decided upon in advance of the drawing. 

The same rule follows us into the logic of induction 
and hypothesis. If in sampling any class, say the M s, 
we first decide what the character P is for which we 
propose to sample that class, and also how many instan 
ces we propose to draw, our inference is really made 
before these latter are drawn, that the proportion of P s 
in the whole class is probably about the same as among 
the instances that are to be drawn, and the only thing 
we have to do is to draw them and observe the ratio. 
But suppose we were to draw our inferences without 
the predesignation of the character P; then we might in 
every case find some recondite character in which those 
instances would all agree. That, by the exercise of 
sufficient ingenuity, we should be sure to be able to do 
this, even if not a single other object of the class M 
possessed that character, is a matter of demonstration. 
For in geometry a curve may be drawn through any 
given series of points, without passing through any one 
of another given series of points, and this irrespective of 
the number of dimensions. Now, all the qualities of 
objects may be conceived to result from variations of a 
number of continuous variables ; hence any lot of ob 
jects possesses some character in common, not possessed 
by any other. It is true that if the universe of quality 



A THEORY OF PROBABLE INFERENCE. 163 

is limited, this is not altogether true ; but it remains 
true that unless we have some special premise from 
which to infer the contrary, it always may be possible 
to assign some common character of the instances , S", 
S ", etc., drawn at random from among the M s, which 
does not belong to the M a generally. So that if the 
character P were not predesignate, the deduction of 
which our induction is the apagogical inversion would 
not be valid ; that is to say, we could not reason that if 
the M B did not generally possess the character P, it 
would not be likely that the s should all possess this 
character. 

I take from a biographical dictionary the first five 
names of poets, with their ages at death. They are, 

Aagard, died at 48. 

Abeille," " " 76. 

Abulola, " 84. 

Abunowas, " " 48. 

Accords, " " 45. 

These five ages have the following characters in com 
mon : 

1. The difference of the two digits composing the 
number, divided by three, leaves a remainder of one. 

2. The first digit raised to the power indicated by the 
second, and then divided by three, leaves a remainder of 
one. 

3. The sum of the prime factors of each age, including 
one as a prime factor, is divisible by three. 

Yet there is not the smallest reason to believe that the 
next poet s age would possess these characters. 

Here we have a conditio sine qud non of valid induc 
tion which has been singularly overlooked by those who 
have treated of the logic of the subject, and is very fre- 



164 A THEORY OF PEOBABLE INFERENCE. 

quently violated by those who draw inductions. So ac 
complished a reasoner as Dr. Lyon Playfair, for instance, 
has written a paper of which the following is an abstract. 
He first takes the specific gravities of the three allotropic 
forms of carbon, as follows : 

Diamond, 3.48 
Graphite, 2.29 
Charcoal, 1.88 

He now seeks to find a uniformity connecting these three 
instances; and he discovers that the atomic weight of 
carbon, being 12, 

Sp. gr. diamond nearly = 3.46 



" " graphite " = 2.29 = y!2 
" " charcoal = 1.86 = j/12 

This, he thinks, renders it probable that the specific 
gravities of the allotropic forms of other elements would, 
if we knew them, be found to equal the different roots of 
their atomic weight. But so far, the character in which 
the instances agree not having been predesignated, the 
induction can serve only to suggest a question, and ought 
not to create any belief. To test the proposed law, he 
selects the instance of silicon, which like carbon exists 
in a diamond and in a graphitoidal condition. He finds 
for the specific gravities 

Diamond silicon, 2.47 
. Graphite silicon, 2.33.* 

* The author ought to have noted that this number is open to some 
doubt, since the specific gravity of this form of silicon appears to vary 
largely. If a different value had suited the theory better, he might have 
been able to find reasons for preferring that other value. But I do not 
mean to imply that Dr. Playfair has not dealt with perfect fairness with 
his facts, except as to the fallacy which I point out. 



A THEORY OF PROBABLE INFERENCE. 165 

Now, the atomic weight of silicon, that of carbon being 
12, can only be taken as 28. But 2.47 does not approx 
imate to any root of 28. It is, however, nearly the 
cube root of 14, (<\X-i- X 28 = 2.41), while 2.33 is nearly 
the fourth root of 28 (v"28 = 2.30). Dr. Playfair claims 
that silicon is an instance satisfying his formula. But 
in fact this instance requires the formula to be modified ; 
and the modification not being predesignate, the instance 
cannot count. Boron also exists in a diamond and a 
graphitoidal form ; and accordingly Dr. Playfair takes 
this as his next example. Its atomic weight is 10.9, and 
its specific gravity is 2.68 ; which is the square root of 
f X 10.9. There seems to be here a further modification 
of the formula not predesignated, and therefore this in 
stance can hardly be reckoned as confirmatory. The 
next instances which would occur to the mind of any 
chemist would be phosphorus and sulphur, which exist 
in familiarly known allotropic forms. Dr. Playfair ad 
mits that the specific gravities of phosphorus have no 
relations to its atomic weight at all analogous to those 
of carbon. The different forms of sulphur have nearly 
the same specific gravity, being approximately the fifth 
root of the atomic weight 32. Selenium also has two 
.allotropic forms, whose specific gravities are 4.8 and 4.3 ; 
one of these follows the law, while the other does not. 
For tellurium the law fails altogether ; but for bromine 
and iodine it holds. Thus the number of specific gravi 
ties for which the law was predesignate are 8 ; namely, 
2 for phosphorus, 1 for sulphur, 2 for selenium, 1 for 
tellurium, 1 for bromine, and 1 for iodine. The law 
holds for 4 of these, and the proper inference is that 
about half the specific gravities of metalloids are roots 
of some simple ratio of their atomic weights. 

Having thus determined this ratio, we proceed to 



166 A THEORY OF PEOBABLE INFERENCE. 

inquire whether an agreement half the time with the 
formula constitutes any special connection between the 
specific gravity and the atomic weight of a metalloid. 
As a test of this, let us arrange the elements in the order 
of their atomic weights, and compare the specific gravity 
of the first with the atomic weight of the last, that of 
the second with the atomic weight of the last but one, 
and so on. The atomic weights are 

Boron, 10.9 Tellurium, 128.1 

Carbon, 12.0 Iodine, 126.9 

Silicon, 28.0 Bromine, 80.0 

Phosphorus, 31.0 Selenium, 79.1 

Sulphur, 32. 

There are three specific gravities given for carbon, and 
two each for silicon, phosphorus, and selenium. The 
question, therefore, is, whether of the fourteen specific 
gravities as many as seven are in Playfair s relation 
with the atomic weights, not of the same element, but 
of the one paired with it. Now, taking the original 
formula of Playfair we find 





Sp. 


gr. 


boron 


= 2.68 


^Te 


= 2 


.64 


3 d 


Sp. 


gr. 


carbon 


= 1.88 


/ V /I 


= 1 


.84 


2 d 


Sp. 


gr. 


carbon 


= 2.29 


V* 


= 2 


.24 


1 st 


Sp. 


gr. 


phosphorus 


= 1.83 


^/Se 


= 1 


.87 


2 d 


Sp. 


gr. 


phosphorus 


= 2.10 


V /Se 


= 2 


.07 



or five such relations without counting that of sulphur 
to itself. Next, with the modification introduced by Play- 
fair, we have 

1 st Sp. gr. silicon = 2.47 $% X Br = 2.51 

2 d Sp. gr. silicon = 2.33 A/2 X Br = 2.33 

Sp. gr. iodine = 4.95 ^2x0= 4.90 

1 st Sp. gr. carbon = 3.48 ^ X I = 3.48 



A THEORY OF PROBABLE INFERENCE. 167 

It thus appears that there is no more frequent agree 
ment with Playfair s proposed law than what is due to 
chance. 1 

Another example of this fallacy was " Bode s law " of 
the relative distances of the planets, which was shattered 
by the first discovery of a true planet after its enuncia 
tion. In fact, this false kind of induction is extremely 
common in science and in medicine. 2 In the case of 
hypothesis, the correct rule has often been laid down ; 
namely, that a hypothesis can only be received upon the 
ground of its having been verified by successful prediction. 
The term predesignation used in this paper appears to be 
more exact, inasmuch as it is not at all requisite that the 
ratio p should be given in advance of the examination of 
the samples. Still, since p is equal to 1 in all ordinary 
hypotheses, there can be no doubt that the rule of pre 
diction, so far as it goes, coincides with that here laid 
down. 

We have now to consider an important modification of 
the rule. Suppose that, before sampling a class of objects, 
we have predesignated not a single character but n char 
acters, for which we propose to examine the samples. 
This is equivalent to making n different inductions from 
the same instances. The probable error in this case is 
that error whose probability for a simple induction is only 
(|) n , and the theory of probabilities shows that it in- 

1 As the relations of the different powers of the specific gravity would 
be entirely different if any other substance than water were assumed as 
the standard, the law is antecedently in the highest degree improbable. 
This makes it likely that some fallacy was committed, but does not show 
what it was. 

2 The physicians seem to use the maxim that you cannot reason from 
post hoc to propter hoc to mean (rather obscurely) that cases must not be 
used to prove a proposition that has only been suggested by these cases 
themselves. 



168 A THEOEY OF PEOBABLE INFEEENCE. 

creases but slowly with n ; in fact, for n 1000 it is only 
about five times as great as for n = 1, so that with only 
25 times as many instances the inference would be as 
secure for the former value of n as with the latter ; with 
100 times as many instances an induction in which n 
10,000,000,000 would be equally secure. Now the whole 
universe of characters will never contain such a number 
as the last ; and the same may be said of the universe of 
objects in the case of hypothesis. So that, without any 
voluntary predesignation, the limitation of our imagina 
tion and experience amounts to a predesignation far 
within those limits ; and we thus see that if the number 
of instances be very great indeed, the failure to predes- 
ignate is not an important fault. Of characters at all 
striking, or of objects at all familiar, the number will 
seldom reach 1,000 ; and of very striking characters or 
very familiar objects the number is still less. So that if 
a large number of samples of a class are found to have 
some very striking character in common, or if a large 
number of characters of one object are found to be pos 
sessed by a very familiar object, we need not hesitate to 
infer, in the first case, that the same characters belong 
to the whole class, or, in the second case, that the two 
objects are practically identical ; remembering only that 
the inference is less to be relied upon than it would be 
had a deliberate predesignation been made. This is no 
doubt the precise significance of the rule sometimes laid 
down, that a hypothesis ought to be simple, simple 
here being taken in the sense of familiar. 

This modification of the rule shows that, even in the 
absence of voluntary predesignation, some slight weight 
is to be attached to an induction or hypothesis. And 
perhaps when the number of instances is not very small, 
it is enough to make it worth while to subject the in- 



A THEORY OF PROBABLE INFERENCE. 169 

ference to a regular test. But our natural tendency will 
be to attach too much importance to sucli suggestions, 
and we shall avoid waste of time in passing them by 
without notice until some stronger plausibility presents 
itself. 

IX. 

In almost every case in which we make an induction 
or a hypothesis, we have some knowledge which renders 
our conclusion antecedently likely or unlikely. The ef 
fect of such knowledge is very obvious, and needs no 
remark. But what also very often happens is that we 
have some knowledge, which, though not of itself bearing 
upon the conclusion of the scientific argument, yet serves 
to render our inference more or less probable, or even 
to alter the terms of it. Suppose, for example, that we 
antecedently know that all the M s strongly resemble 
one another in regard to characters of a certain order. 
Then, if we find that a moderate number of M 9 s taken 
at random have a certain character, P, of that order, we 
shall attach a greater weight to the induction than we 
should do if we had not that antecedent knowledge. 
Thus, if we find that a certain sample of gold has a 
certain chemical character, since we have very strong 
reason for thinking that all gold is alike in its chemical 
characters, we shall have no hesitation in extending 
the proposition from the one sample to gold in general. 
Or if we know that among a certain people, say the 
Icelanders, an extreme uniformity prevails in regard 
to all their ideas, then, if we find that two or three in 
dividuals taken at random from among them have all 
any particular superstition, we shall be the more ready 
to infer that it belongs to the whole people from what 
we know of their uniformity. The influence of this sort 



170 A THEORY OF PKOBABLE INFERENCE. 

of uniformity upon inductive conclusions was strongly in 
sisted upon by Philodemus, and some very exact concep 
tions in regard to it may be gathered from the writings 
of Mr. Galton. Again, suppose we know of a certain 
character, P, that in whatever classes of a certain des 
cription it is found at all, to those it usually belongs as 
a universal character ; then any induction which goes 
toward showing that all the M s are P will be greatly 
strengthened. Thus it is enough to find that two or 
three individuals taken at random from a genus of ani 
mals have three toes on each foot, to prove that the same 
is true of the whole genus ; for we know that this is a 
generic character. On the other hand, we shall be slow 
to infer that all the animals of a genus have the same 
color, because color varies in almost every genus. This 
kind of uniformity seemed to J. S. Mill to have so con 
trolling an influence upon inductions, that he has taken 
it as the centre of his whole theory of the subject. 

Analogous considerations modify our hypothetic infer 
ences. The sight of two or three words will be sufficient 
to convince me that a certain manuscript was written by 
myself, because I know a certain look is peculiar to it. 
So an analytical chemist, who wishes to know whether a 
solution contains gold, will be completely satisfied if it 
gives a precipitate of the purple of cassius with chloride 
of tin ; because this proves that either gold or some hith 
erto unknown substance is present. These are examples 
of characteristic tests. Again, we may know of a certain 
person, that whatever opinions he holds he carries out 
with uncompromising rigor to their utmost logical con 
sequences ; then, -if we find his views bear some of the 
marks of any ultra school of thought, we shall readily 
conclude that he fully adheres to that school. 

There are thus four different kinds of uniformity and 



A THEORY OF PROBABLE INFERENCE. 171 

non-uniformity which may influence our ampliative in 
ferences : 

1. The members of a class may present a greater or 
less general resemblance as regards a certain line of char 
acters. 

2. A character may have a greater or less tendency 
to be present or absent throughout the whole of whatever 
classes of certain kinds. 

3. A certain set of characters may be more or less 
intimately connected, so as to be probably either present 
or absent together in certain kinds of objects. 

4. An object may have more or less tendency to 
possess the whole of certain sets of characters when it 
possesses any of them. 

A consideration of this sort may be so strong as to 
amount to demonstration of the conclusion. In this case, 
the inference is mere deduction, that is, the application 
of a general rule already established. In other cases, the 
consideration of uniformities will not wholly destroy the 
inductive or hypothetic character of the inference, but 
will only strengthen or weaken it by the addition of a 
new argument of a deductive kind. 



X. 

We have thus seen how, in a general way, the processes 
of inductive and hypothetic inference are able to afford 
answers to our questions, though these may relate to 
matters beyond our immediate ken. In short, a theory 
of the logic of verification has been sketched out. This 
theory will have to meet the objections of two opposing 
schools of logic. 

The first of these explains induction by what is called 
the doctrine of Inverse Probabilities, of which the follow- 



172 A THEORY OF PROBABLE INFERENCE. 

ing is an example : Suppose an ancient denizen of the 
Mediterranean coast, who had never heard of the tides, 
had wandered to the shore of the Atlantic Ocean, and 
there, on a certain number m of successive days had 
witnessed the rise of the sea. Then, says Quetelet, he 
would have been entitled to conclude that there was a 
probability equal to ^ t_ that the sea would rise on the 
next following day. 1 Putting m = 0, it is seen that 
this view assumes that the probability of a totally un 
known event is ; or that of all theories proposed for 
examination one half are 4;rue. In point of fact, we 
know that although theories are not proposed unless 
they present some decided plausibility, nothing like one 
half turn out to be true. But to apply correctly the 
doctrine of inverse probabilities, it is necessary to know 
the antecedent probability of the event whose proba 
bility is in question. Now, in pure hypothesis or induc 
tion, we know nothing of the conclusion antecedently 
to the inference in hand. Mere ignorance, however, 
cannot advance us toward any knowledge ; therefore it 
is impossible that the theory of inverse probabilities 
should rightly give a value for the probability of a pure 
inductive or hypothetic conclusion. For it cannot do 
this without assigning an antecedent probability to this 
conclusion ; so that if this antecedent probability rep 
resents mere ignorance (which never aids us), it cannot 
do it at all. 

The principle which is usually assumed by those who 
seek to reduce inductive reasoning to a problem in in 
verse probabilities is, that if nothing whatever is known 
about the frequency of occurrence of an event, then any 
one frequency is as probable as any other. But Boole 

1 See Laplace, "Theorie Analitique des Probabilites," livre ii. chap. vi. 



A THEORY OF PROBABLE INFERENCE. 173 

has shown that there is no reason whatever to prefer this 
assumption, to saying that any one " constitution of the 
universe" is as probable as any other. Suppose, for 
instance, there were four possible occasions upon which 
an event might occur. Then there would be 16 " con 
stitutions of the universe," or possible distributions of 
occurrences and non-occurrences. They are shown in 
the following table, where Y stands for an occurrence 
and N for a non-occurrence. 



4 occurrences. 


3 occurrences. 


2 occurrences. 


1 occurrence. 


occurrence. 


YYYY 


YYYN 


YYNN 


YNNN 


NNNN 




YYNY 


YNYN 


NYNN 






YNYY 


YNNY 


NNYN 






NYYY 


NYYN 


NNNT 








NYNY 







NNYY 

It will be seen that different frequencies result some 
from more and some from fewer different " constitutions 
of the universe," so that it is a very different thing to 
assume that all frequencies are equally probable from 
what it is to assume that all constitutions of the universe 
are equally probable. 

Boole says that one assumption is as good as the other. 
But I will go further, and say that the assumption that 
all constitutions of the universe are equally probable is 
far better than the assumption that all frequencies are 
equally probable. For the latter proposition, though it 
may be applied to any one unknown event, cannot be 
applied to all unknown events without inconsistency. 
Thus, suppose all frequencies of the event whose occur 
rence is represented by I^in the above table are equally 
probable. Then consider the event which consists in a 
Y following a Y or an N following an N. The possible 



174 A THEORY OF PROBABLE INFERENCE. 

ways in which this event may occur or not are shown in 
the following table : 



3 occurrences. 

YYYY 

NNNN 


2 occurrences, 

YYYN 

NNNY 


1 occurrence. 

YYNY 

NNYN 


occurrence. 

YNYN 
NYNY 


YYNN 
NNYY 


YNNY 
NYYN 


N YYY 

YNNN 


YNYY 

NYNN 



It will be found that assuming the different frequencies 
of the first event to be equally probable, those of this new 
event are not so, the probability of three occurrences 
being half as large again as that of two, or one. On the 
other hand, if all constitutions of the universe are equally 
probable in the one case, they are so in the other ; and 
this latter assumption, in regard to perfectly unknown 
events, never gives rise to any inconsistency. 

Suppose, then, that we adopt the assumption that any 
one constitution of the universe is as probable as any 
other ; how will the inductive inference then appear, con 
sidered as a problem in probabilities ? The answer is 
extremely easy ; 1 namely, the occurrences or non-occur 
rences of an event in the past in no way affect the proba 
bility of its occurrence in the future. 

Boole frequently finds a problem in probabilities to be 
indeterminate. There are those to whom the idea of an 
unknown probability seems an absurdity. Probability, 
they say, measures the state of our knowledge, and ig 
norance is denoted by the probability |. But I appre 
hend that the expression " the probability of an event " 
is an incomplete one. A probability is a fraction whose 

1 See Boole, "Laws of Thought." 



A THEORY OF PROBABLE INFERENCE. 175 

numerator is the frequency of a specific kind of event, 
while its denominator is the frequency of a genus embrac 
ing that species. Now the expression in question names 
the numerator of the fraction, but omits to name the de 
nominator. There is a sense in which it is true that the 
probability of a perfectly unknown event is one half ; 
namely, the assertion of its occurrence is the answer to 
a possible question answerable by " yes " or " no," and 
of all such questions just half the possible answers are 
true. But if attention be paid to the denominators of 
the fractions, it will be found that this value of J is one 
of which no possible use can be made in the calculation 
of probabilities. 

The theory here proposed does not assign any proba 
bility to the inductive or hypothetic conclusion, in the 
sense of undertaking to say how frequently that conclu 
sion would be found true. It does not propose to look 
through all the possible universes, and say in what pro 
portion of them a certain uniformity occurs ; such a 
proceeding, were it possible, would be quite idle. The 
theory here presented only says how frequently, in this 
universe, the special form of induction or hypothesis 
would lead us right. The probability given by this theory 
is in every way different in meaning, numerical value, 
and form from that of those who would apply to am- 
pliative inference the doctrine of inverse chances. 

Other logicians hold that if inductive and hypothetic 
premises lead to true oftener than to false conclusions, 
it is only because the universe happens to have a certain 
constitution. Mill and his followers maintain that there 
is a general tendency toward uniformity in the universe, 
as well as special uniformities such as those which we 
have considered. The Abbe* Gratry believes that the 
tendency toward the truth in induction is due to a mirac- 



176 A THEORY OF PROBABLE INFERENCE. 

ulous intervention of Almighty God, whereby we are led 
to make such inductions as happen to be true, and are 
prevented from making those which are false. Others 
have supposed that there is a special adaptation of the 
mind to the universe, so % that we are more apt to make 
true theories than we otherwise should be. Now, to say 
that a theory such as these is necessary to explaining the 
validity of induction and hypothesis is to say that these 
modes of inference are not in themselves valid, but that 
their conclusions are rendered probable by being probable 
deductive inferences from a suppressed (and originally 
unknown) premise. But I maintain that it has been 
shown that the modes of inference in question are neces 
sarily valid, whatever the constitution of the universe, so 
long as it admits of the premises being true. Yet I am 
willing to concede, in order to concede as much as possi 
ble, that when a man draws instances at random, all that 
he knows is that he tries to follow a certain precept ; so 
that the sampling process might be rendered generally 
fallacious by the existence of a mysterious and malign 
connection between the mind and the universe, such that 
the possession by an object of an unperceived character 
might influence the will toward choosing it or rejecting 
it. Such a circumstance would, however, be as fatal to 
deductive as to ampliative inference. Suppose, for exam 
ple, that I were to enter a great hall where people were 
playing rouge et noir at many tables ; and suppose that 
I knew that the red and black were turned up with equal 
frequency. Then, if I were to make a large number of 
mental bets with myself, at this table and at that. I. might, 
by statistical deduction, expect to win about half of them, 
precisely as I might expect, from the results of these 
samples, to infer by induction the probable ratio of fre 
quency of the turnings of red and black in the long run, 



A THEORY OF PROBABLE INFERENCE. 177 

if I did not know it. But could some devil look at eacli 
card before it was turned, and then influence me mentally 
to bet upon it or to refrain therefrom, the observed ratio 
in the cases upon which I had bet might be quite different 
from the observed ratio in those cases upon which I had 
not bet. I grant, then, that even upon my theory some 
fact has to be supposed to make induction and hypothe 
sis valid processes ; namely, it is supposed that the su 
pernal powers withhold their hands and let me alone, 
and that no mysterious uniformity or adaptation inter 
feres with the action of chance. But then this negative 
fact supposed by my theory plays a totally different part 
from the facts supposed to be requisite by the logicians 
of whom I have been speaking. So far as facts like those 
they suppose can have any bearing, they serve as major 
premises from which the fact inferred by induction or 
hypothesis might be deduced ; while the negative fact 
supposed by me is merely the denial of any major premise 
from which the falsity of the inductive or hypothetic con 
clusion could in general be deduced. Nor is it necessary 
to deny altogether the existence of mysterious influences 
adverse to the validity of the inductive and hypothetic 
processes. So long as their influence were not too over 
whelming, the wonderful self-correcting nature of the 
ampliative inference would enable us, even if they did 
exist, to detect and make allowance for them. 

Although the universe need have no peculiar consti 
tution to render ampliative inference valid, yet it is worth 
while to inquire whether or not it has such a constitu 
tion ; for if it has, that circumstance must have its effect 
upon all our inferences. It cannot any longer be denied 
that the human intellect is peculiarly adapted to the 
comprehension of the laws and facts of nature, or at 
least of some of them ; and the effect of this adaptation 



178 A THEORY OF PROBABLE INFERENCE. 

upon our reasoning will be briefly considered in the next 
section. Of any miraculous interference by the higher 
powers, we know absolutely nothing ; and it seems in 
the present state of science altogether improbable. The 
effect of a knowledge of special uniformities upon ampli- 
ative inferences has already been touched upon. That 
there is a general tendency toward uniformity in nature 
is not merely an unfounded, it is an absolutely absurd, 
idea in any other sense than that man is adapted to his 
surroundings. For the universe of marks is only limited 
by the limitation of human interests and powers of ob 
servation. Except for that limitation, every lot of objects 
in the universe would have (as I have elsewhere shown) 
some character in common and peculiar to it. Conse 
quently, there is but one possible arrangement of charac 
ters among objects as they exist, and there is no room 
for a greater or less degree of uniformity in nature. If 
nature seems highly uniform to us, it is only because our 
powers are adapted to our desires. 

XI. 

The questions discussed in this essay relate to but a 
small part of the Logic of Scientific Investigation. Let 
us just glance at a few of the others. 

Suppose a being, from some remote part of the uni 
verse, where the conditions of existence are inconceivably 
different from ours, to be presented with a United States 
Census Report, which is for us a mine of valuable in 
ductions, so vast as almost to give that epithet a new signi 
fication. He begins, perhaps, by comparing the ratio of 
indebtedness to deaths by consumption in counties whose 
names begin with the different letters of the alphabet. 
It is safe to say that he would find the ratio everywhere 



B a i T i s 




WINTER KA1N FATX 

CASchott 



A THEOKY OF PROBABLE INFERENCE. 179 



the same, and thus his inquiry would lead to nothing. 
For an induction is wholly unimportant unless the pro 
portions of P s among the M s and among the non-M s 
differ ; and a hypothetic inference is unimportant unless 
it be found that S has either a greater or a less propor 
tion of the characters of M than it has of other charac 
ters. The stranger to this planet might go on for some 
time asking inductive questions that the Census would 
faithfully answer, without learning anything except that 
certain conditions were independent of others. At length, 
it might occur to him to compare the January rain-fall 
with the illiteracy. What he would find is given in the 
folio win": table 1 : 



REGION. 


January Rain-fall. 


Illiteracy. 


Atlantic Sea-coast, Port-) 
land to Washington ) 


Inches. 

0.92 


Per cent. 
11 


Vermont, Northern and) 
Western New York ) 


0.78 


7 


Upper Mississippi E/iver . 


0.52 


3 


Ohio River Valley . . . 


0.74 


8 


Lower Mississippi, Red) 
River, and Kentucky ) 


1.08 


50 


Mississippi Delta and) 
Northern Gulf Coast ) 


1.09 


57 


Southeastern Coast . . . 


0.68 


40 



1 The different regions with the January rain-fall are taken from Mr. 
Schott s work. The percentage of illiteracy is roughly estimated from the 
numbers given in the Keport of the 1870 Census. 



180 A THEORY OF PROBABLE INFERENCE. 

He would infer that in places that are drier in January 
there is, not always but generally, less illiteracy than 
in wetter places. A detailed comparison between Mr. 
Schott s map of the winter rain-fall with the map of 
illiteracy in the general census, would confirm the result 
that these two conditions have a partial connection. 
This is a very good example of an induction in which 
the proportion of P s among the M 9 s is different, but 
not very different, from the proportion among the non- 
Jf s. It is unsatisfactory ; it provokes further inquiry ; 
we desire to replace the M by some different class, so 
that the two proportions may be more widely separated. 
Now we, knowing as much as we do of the effects of 
winter rain-fall upon agriculture, upon wealth, etc., and 
of the causes of illiteracy, should come to such an inquiry 
furnished with a large number of appropriate conceptions ; 
so that we should be able to ask intelligent questions not 
unlikely to furnish the desired key to the problem. But 
the strange being we have imagined could only make his 
inquiries hap-hazard, and could hardly hope ever to find 
the induction of which he was in search. 

Nature is a far vaster and less clearly arranged reper 
tory of facts than a census report ; and if men had not 
come to it with special aptitudes for guessing right, it 
may well be doubted whether in the ten or twenty thou 
sand years that they may have existed their greatest 
mind would have attained the amount of knowledge 
which is actually possessed by the lowest idiot. But, 
in point of fact, not man merely, but all animals derive 
by inheritance (presumably by natural selection) two 
classes of ideas which adapt them to their environment. 
In the first place, they all have from. birth some notions, 
however crude and concrete, of force, matter, space, and 
time ; and, in the next place, they have some notion of 



A THEORY OF PROBABLE INFERENCE. 181 

what sort of objects their fellow-beings are, and of how 
they will act on given occasions. Our innate mechanical 
ideas were so nearly correct that they needed but slight 
correction. The fundamental principles of statics were 
made out by Archimedes. Centuries later Galileo began 
to understand the laws of dynamics, which in our times 
have been at length, perhaps, completely mastered. The 
other physical sciences are the results of inquiry based 
on guesses suggested by the ideas of mechanics. The 
moral sciences, so far as they can be called sciences, 
are equally developed out of our instinctive ideas about 
human nature. Man has thus far not attained to any 
knowledge that is not in a wide sense either mechanical 
or anthropological in its nature, and it may be reasonably 
presumed that he never will. 

Side by side, then, with the well established propo 
sition that all knowledge is based on experience, and 
that science is only advanced by the experimental verifi 
cations of theories, we have to place this other equally 
important truth, that all human knowledge, up to the 
highest flights of science, is but the development of our 
inborn animal instincts. 



NOTE A. 



BOOLE, De Morgan, and their followers, frequently 
speak of a " limited universe of discourse " in logic. An 
unlimited universe would comprise the whole realm of the 
logically possible. In such a universe, every universal 
proposition, not tautologous, is false ; every particular 
proposition, not absurd, is true. Our discourse seldom 
relates to this universe : we are either thinking of the 
physically possible, or of the historically existent, or of 
the world of some romance, or of some other limited 
universe. 

But besides its universe of objects, our discourse also 
refers to a universe of characters. Thus, we might 
naturally say that virtue and an orange have nothing 
in common. It is true that the English word for each 
is spelt with six letters, but this is not one of the marks 
of the universe of our discourse. 

A universe of things is unlimited in which every com 
bination of characters, short of the whole universe of 
characters, occurs in some object. In like manner, the 
universe of characters is unlimited in case every aggre 
gate of things short of the whole universe of things 
possesses in common one of the characters of the uni 
verse of characters. The conception of ordinar}^ syllo 
gistic is so unclear that it would hardly be accurate to 
say that it supposes an unlimited universe of characters ; 



ON A LIMITED UNIVERSE OF MARKS. 183 

but it comes nearer to that than to any other consistent 
view. The non-possession of any character is regarded 
as implying the possession of another character the nega 
tive of the first. 

In our ordinary discourse, on the other hand, not only 
are both universes limited, but, further than that, we 
have nothing to do with individual objects nor simple 
marks ; so that we have simply the two distinct universes 
of things and marks related to one another, in general, in 
a perfectly indeterminate manner. The consequence is, 4 
that a proposition concerning the relations of two groups 
of marks is not necessarily equivalent to any proposition 
concerning classes of things ; so that the distinction 
between propositions in extension and propositions in 
comprehension is a real one, separating two kinds of 
facts, whereas in the view of ordinary syllogistic the 
distinction only relates to two modes of considering any 
fact. To say that every object of the class S is included 
among the class of P s, of course must imply that every 
common character of the P s is a common character of 
the $ s. But the converse implication is by no means 
necessary, except with an unlimited universe of marks. 
The reasonings in depth of which I have spoken, suppose, 
of course, the absence of any general regularity about the 
relations of marks and things. 

I may mention here another respect in which this view 
differs from that of ordinary logic, although it is a point 
which has, so far as I am aware, no bearing upon the 
theory of probable inference. It is that under this view 
there are propositions of which the subject is a class of 
things, while the predicate is a group of marks. Of such 
propositions there are twelve species, distinct from one 
another in the sense that any fact capable of being ex 
pressed by a proposition of one of these species cannot 



184 ON A LIMITED UNIVERSE OF MARKS. 

be expressed by any proposition of another species. The 
following are examples of six of the twelve species : 

1. Every object of the class S possesses every character of 
the group TT. 

2. Some object of the class S possesses all characters of 
the group TT. 

3. Every character of the group TT is possessed by some 
object of the class S. 

4. Some character of the group TT is possessed by all the 
objects of the class S. 

5. Every object of the class S possesses some character of 
the group TT. 

6. Some object of the class S possesses some character of 
the group TT. 

The remaining six species of propositions are like the 
above, except that they speak of objects wanting charac 
ters instead of possessing characters. 

But the varieties of proposition do not end here ; for 
we may have, for example, such a form as this : " Some 
object of the class S possesses every character not want 
ing to any object of the class P." In short, the relative 
term " possessing as a character," or its negative, may 
enter into the proposition any number of times. We 
may term this number the order of the proposition. 

An important characteristic of this kind of logic is the 
part that immediate inference plays in it. Thus, the 
proposition numbered 3, above, follows from No. 2, and 
No. 5 from No. 4. It will be observed that in both cases 
a universal proposition (or one that states the non- 
existence of something) follows from a particular propo 
sition (or one that states the existence of something). 
All the immediate inferences are essentially of that 
nature. A particular proposition is never immediately 
inferable from a universal one. (It is true that from 



ON A LIMITED UNIVERSE OF MARKS. 185 

" no A exists " we can infer that " something not A 
exists ; " but this is not properly an immediate infer 
ence, it really supposes the additional premise that 
u something exists.") There are also immediate in 
ferences raising and reducing the order of propositions. 
Thus, the proposition of the second order given in the 
last paragraph follows from " some S is a P." On the 
other hand, the inference holds, 

Some common character of the S s is wanting to every 
thing except _P s ; 
. . Every S is a P. 

The necessary and sufficient condition of the existence 
of a syllogistic conclusion from two premises is simple 
enough. There is a conclusion if, and only if, there is 
a middle term distributed in one premise and undistribu 
ted in the other. But the conclusion is of the kind called 
spurious l by De Morgan if, and only if, the middle term 
is affe cted by a " some " in both premises. For exam 
ple, let the two premises be, 

Every object of the class /S wants some character of the 
group p.; 

Every object of the class P possesses some character not of 
the group //,. 

The middle term /JL is distributed in the second premise, 
but not in the first ; so that a conclusion can be drawn. 
But, though both propositions are universal, ^ is under 
a " some " in both ; hence only a spurious conclusion 
can be drawn, and in point of fact we can infer both of 
the following : 

1 On spurious propositions, see Mr. B. I. Oilman s paper in the Johns 
Hopkins University Circular for August, 1882. The number of such 
forms in any order is probably finite. 



186 ON A LIMITED UNIVERSE OF MARKS. 

Every object of the class S wants a character other than 
some character common to the class P ; 

Every object of the class P possesses a character other 
than some character wanting to every object of the class S. 

The order of the conclusion is always the sum of the 
orders of the premises ; but to draw up a rule to deter 
mine precisely what the conclusion is, would be difficult. 
It would at the same time be useless, because the prob 
lem is extremely simple when considered in the light of 
the logic of relatives. 



NOTE B. 



A DUAL relative term, such as " lover," " benefactor," 
" servant," is a common name signifying a pair of ob 
jects. Of the two members of the pair, a determinate 
one is generally the first, and the other the second ; so 
that if the order is reversed, the pair is not considered as 
remaining the same. 

Let A, B, C, D, etc., be all the individual objects in 
the universe ; then all the individual pairs may be arrayed 
in a block, thus : 

A:A A:B A:C A : D etc. 
B:A B:B B:C B : D etc. 
C:A C:B C:C C:D etc. 
D:A D : B D : C D : D etc. 
etc. etc. etc. etc. etc. 

A general relative may be conceived as a logical aggre 
gate of a number of such individual relatives. Let I de 
note " lover ; " then we may write 



where (Z)# is a numerical coefficient, whose value is 1 in 
case I is a lover of J, and in the opposite case, and 
where the sums are to be taken for all individuals in the 



unverse. 



188 THE LOGIC OF EEL ATI YES. 

Every relative term has a negative (like any other 
term) which may be represented by drawing a straight 
line over the sign for the relative itself. The negative 
of a relative includes every pair that the latter excludes, 
and vice versa. Every relative has also a converse, pro 
duced by reversing the order of the members of the pair. 
Thus, the converse of u lover" is "loved." The con 
verse may be represented by drawing a curved line over 
the sign for the relative, thus : I. It is defined by the 
equation 



The following formulae are obvious, but important : 



(i -< b) = (l -< i) (i -< b) = (l-< b). 

Relative terms can be aggregated and compounded like 
others. Using -f for the sign of logical aggregation, and 
the comma for the sign of logical composition (Boole s 
multiplication, here to be called non-relative or internal 
multiplication), we have the definitions 



The first of these equations, however, is to be understood 
in a peculiar way : namely, the + in the second member 
is not strictly addition, but an operation by which 



Instead of (l)$ + (&)y- , we might with more accuracy 
write 





THE LOGIC OF EELATIVES. 189 

The main formulas of aggregation and composition are 

( If I -< s and b -< s, then I + b < s. |_ 
(If s < I and s -< b, then s-< ,&. I 

( If Z + -< 5, then Z -< s and -< s. \ 
(If 5-< Z,#, then s < Z and s -< &. ) 

( (I + &) ?s -< l,s + &,$ ) 
1 (I + s),(b + s) -< l t b + s. ) 

The subsidiary formulas need not be given, being the 
same as in non-relative logic. 

We now come to the combination of relatives. Of 
these, we denote two by special symbols ; namely, we 

write 

lb for lover of a benefactor, 
and 

I f b for lover of everything hut benefactors. 

The former is called a particular combination, because 
it implies the existence of something loved by its relate 
and a benefactor of its correlate. The second combina 
tion is said to be universal, because it implies the non- 
existence of anything except what is either loved by its 
relate or a benefactor of its correlate. The combination 
lb is called a relative product, / f b a relative sum. The 
I and b arc said to be undistributed in both, because if 
I - C s, then lb - C sb and I f b ^< s f b ; and if b -< 5, 
then lb - C Is and I f b -< I f .9. 

The two combinations are defined by the equations 



The sign of addition in the last formula has the same 
signification as in the equation defining non-relative 
multiplication. 



190 THE LOGIC OF KELATIYES. 

Relative addition and multiplication are subject to the 
associative law. That is, 



l(bs) = 

Two formulae so constantly used that hardly anything 
can be done without them are 



The former asserts that whatever is lover of an object 
that is benefactor of everything but a servant, stands to 
everything but servants in the relation of lover of a 
benefactor. The latter asserts that whatever stands to 
any servant in the relation of lover of everything but its 
benefactors, is a lover of everything but benefactors of 
servants. The following formulas are obvious and triv 
ial: 

Is + Is -< (l+b)s 

z,&t-<(*t)(at)- 

Unobvious and important, however, are these : 
(I + b) s -< Is + bs 

(Jt*),(&t*)-<MU 

There are a number of curious development formulae. 
Such are 



(I + b) t s = 2 P {\l f (s 

n (b + s) = 2 P {\_(1 + P ) t b-],[_(l +p) f s-]}. 

The summations and multiplications denoted by ^ and IT 
are to be taken non-relatively, and all relative terms are 
to be successively substituted for p. 



THE LOGIC OF RELATIVES. 191 

The negatives of the combinations follow these rules : 

I I b = Ib I b = I ~f b 

The converses of combinations are as follows : 

Individual dual relatives are of two types, 
A : A and A : B. 

Relatives containing no pair of an object with itself are 
called alio-relatives as opposed to self -relatives. The 
negatives of alio-relatives pair every object with itself. 
Relatives containing no pair of an object with anything 
but itself are called concurrents as opposed to opponents. 
The negatives of concurrents pair every object with every 
other. 

There is but one relative which pairs every object with 
itself and with every other. It is the aggregate of all 
pairs, and is denoted by GO. It is translated into ordi 
nary language by " coexistent with." Its negative is 0. 
There is but one relative which pairs every object with 
itself and none with any other. It is 

(A : A) + (B : B) + (C : C) + etc. ; 

is denoted by 1, and in ordinary language is "identical 
with ." Its negative, denoted by n, is " other than," 
or " not." 

No matter what relative term x may be, we have 

-< x x -< oo. 



192 THE LOGIC OF RELATIVES. 

Hence, obviously 

x + = x x, GO = x 

a?-foo OQ cc , = 0. 

The last formula hold for the relative operations ; thus, 

# f GO := 00 XO = 0. 

GO f sc = oo a? 0. 

The formulas 

X + = X X, GO = X 

also hold if we substitute the relative operations, and 
also 1 for oo, and n for ; thus, 

x f n = x x~L = x. 

n -f x = x \x = x. 
We have also 

l + l=o, 1,1 = 0. 

To these partially correspond the following pair of highly 
important formulas : 

1 -< 1 1 1 l~l -< n. 

The logic of relatives is highly multiform ; it is char 
acterized by innumerable immediate inferences, and by 
various distinct conclusions from the same sets of premi 
ses. An example of the first character is afforded by 
Mr. Mitchell s F lv following from F lv ,. As an instance 
of the second, take the premises, 

Every man is a lover of an animal ; 
and 

Every woman is a lover of a non-animal. 

From these we can equally infer that 

Every man is a lover of something which stands to each 
woman in the relation of not being the only thing loved 
by her, 



THE LOGIC OF RELATIVES. 193 



and that 



Every woman is a lover of something which stands to 
each man in the relation of not being the only thing loved 
by him. 

The effect of these peculiarities is that this algebra can 
not be subjected to hard and fast rules like those of 
the Boolian calculus ; and all that can be done in this 
place is to give a general idea of the way of working with 
it. The student must at the outset disabuse himself of 
the notion that the chief instruments of algebra are the 
inverse operations. General algebra hardly knows any 
inverse operations. When an inverse operation is iden 
tical with a direct operation with an inverse quantity 
(as subtraction is the addition of the negative, and as 
division is multiplication by the reciprocal), it is useful ; 
otherwise it is almost always useless. In ordinary alge 
bra, we speak of the " principal value " of the logarithm, 
etc., which is a direct operation substituted for an in 
definitely ambiguous inverse operation. The elimination 
and transposition in this algebra really does depend, 
however, upon formulae quite analogous to the 

x + (- x) = x X \ = 1, 

of arithmetical algebra. These formulas are 



I + 1 = oo 1 -< 1 1 . 

For example, to eliminate * from the two propositions 
1-C Is l-< sb, 

we relatively multiply them in such an order as to bring 
the two s s together, and then apply the second of the 
above formulas, thus : 

1 -< IsSb -< l\\b. 



194 THE LOGIC OF EELATIVES. 

This example shows the use of the association formulae 
in bringing letters together. Other formulas of great 
importance for this purpose are 



The distribution formula are also useful for this pur 
pose. 

When the letter to be eliminated has thus been re 
placed by one of the four relatives, 0, GO, 1, n, the 
replacing relative can often be got rid of by means of 
one of the formulae 



When we have only to deal with universal propositions, 

.it will be found convenient so to transpose everything 

from subject to predicate as to make the subject L Thus, 

if we have given I -< 6, we may relatively add I to both 

sides ; whereupon we have 



Every proposition will then be in one of the forms 
1 -< b 1 1 l-<bl. 



With a proposition of the form 1 <^ b f ?, we have the 
right (1) to transpose the terms, and (2) to convert the 
terms. Thus, the following are equivalent : 

1 -< b 1 1 



1 -< ?t ft- 

With a proposition of the form 1 -< b I, we have only 
the right to convert the predicate giving 1 -< I b. 



THE LOGIC OF RELATIVES. 195 

With three terms, there are four forms of universal 
propositions, namely : 



Of these, the third is an immediate inference from the 
second. 

By way of illustration, we may work out the syllo 
gisms whose premises are the propositions of the first 
order referred to in Note A. Let a and c be class terms, 
and let {3 be a group of characters. Let p he the relative 
" possessing as a character." The non-relative terms 
are to be treated as relatives, a, for instance, being 
considered as " a coexistent with " and a as " coexistent 
with a that is." Then, the six forms of affirmative 
propositions of the first order are 



The various kinds of syllogism are as follows : 
1. Premises : 1 -< a f p f /3 1 -< c ^ p t /? 
Convert one of the premises and multiply, 



The treatment would be the same if one or both of 
the premises were negative ; that is, contained p in place 
of p. 



196 THE LOGIC OF RELATIVES. 

2. Premises : 1 -< a ^p t /? 1 -< c (p f fi). 
We have 



The same with negatives. 

3. Premises : 1 -< a ( p f j3) 1 -< % (p t /?). 

1 -< (JP t P) (t) c -< 
The same with negatives. 

4. Premises : 1 -< f.p f /? 1 -< c 



If one of the premises, say the first, were negative, we 
should obtain a similar conclusion, 



but from this again jt? could be eliminated, giving 
1 -< a f c, or a -< c. 

5. Premises : 1 -< (p t /3) 1 -< (c t^?) y^. 

1 -< o (p t/3)^ (^ t -< -P (^ t c). 
If either premise were negative, ^> could be eliminated, 
giving 1 ^^ 0, or some a is c. 

6. Premises : 1 -< (a t^) /? 1 -< (c ^ p) ft. 



7. Premises : 1 -< a f.P t 

l-< (f^t/5)( 

8. Premises : 1 -< a (p -\ j3) l-<cp1[ 



9. Premises : 1 -< (a f ^) y8 1 -< gp t A 
1 -< (tjP))8 (jSt^c) -< 



THE LOGIC OF RELATIVES. 197 

If one premise is negative, we have the further conclu 
sion 1 <[ dc. 



10. Premises : 1 -< ap f ft 1 -< cp 

1 -< (ap t /?) (j8 t $ c) -< .p t j><?. 

11. Premises : I -< a^p-fft 1 -< 



We might also conclude 



but this conclusion is an immediate inference from the 
other ; for 



If one premise is negative, we have the further conclu 
sion 1 -< a f c. 



12. Premises: 1 -< a (j? | /*) l 
1 -< (^t)8) 08j> t<0 -< 
If one premise is negative, we have the further inference 



13. Premises : l-<(a^p)(3 1 -< f ^ A 

1 -< ( t^) /5 (^ t c) -< (" v t^) (^ t c). 
14 Premises: 1 -< ap-\ ft 1 -< c-fp/3. 



If one premise is negative, we have the further spurious 
inference 1 -<^ a n f <?. 



15. Premises: l-< 

1 -< ( t^ ( t -< tl> (u/> t 
We can al&o infer 1 -< (a f^)^ t c - 



198 THE LOGIC OF RELATIVES. 

16. Premises : 1 -< a ^p f (3 1 -< cpp. 



If one premise is negative, we can further infer 

17. Premises : 1 -< a (p f /?) l-<cpp. 

1 -< a (p t P) fipc -< appc. 
If one premise is negative, we have the further spurious 



conclusion 1 <[ a lie. 

18. Premises: 1 -< (a^p~)p 1 -< 



19. Premises: 1 -< ap} (3 l-<cp(3. 

l-< (ap-tP)jtpc-< appc. 
If one premise is negative, we further conclude 1 

20. Premises : 1 -< a -\p p l-<cpp. 



21. Premises: 1 -< 



When we have to do with particular propositions, we 
have the proposition oo -<[ 0, or "something exists;" 
for every particular proposition implies this. Then every- 
proposition can be put into one or other of the four 
forms 

oo -< 0-j^tO 

00 -< (0 t 00 

oo -< (0 1 1 oo 

OO -< 00 I 00. 

Each of these propositions immediately follows from the 
one above it. The enveloped expressions which form the 



THE LOGIC OF EELATIYES. 199 

predicates have the remarkable property that each is 
either or oo. This fact gives extraordinary freedom 
in the use of the formulas. In particular, since if any 
thing not zero is included under such an expression, the 
whole universe is included, it will be quite unnecessary 
to write the GO -<^ which begins every proposition. 

Suppose that / and g are general relatives signifying 
relations of things to times. Then, Dr. Mitchell s six 
forms of two dimensional propositions appear thus : 



^ ttv = oo/oo. 
It is obvious that I f -< Z, for 

Z|0-< (7-j-O) oo -< ZfO oo-< Jttt-< l 

If then we have Of/fO as one premise, and the other 
contains g, we may substitute for g the product (/, g). 

g -< 0r, oo -< g, (0 f/t 0) -< g,f. 
From the two premises 

oo (/t 0) and f g oo, 
by the application of the formulas 



we have 



These formulae give the first column of Dr. Mitchell s 
rule on page 90. 



200 THE LOGIC OF KELATIVES. 

The following formulae may also be applied 



2. (Ot/)oo 

3. (Of/)oc 

4. (Of/)Gc (0 | S) oc -< (0 t/)yo, 

5. (Ot/tO)(Ot0oo) = Of (#/,/) t 

6. (Ot/)oo (Of #00) = (0t <//,/) oo. 



8. 

9- (Ot/>),(0tflroo) =0f/oo, !7 oo. 

10. (ot/t o)> 000 = ot (/-//,/) to. 

11. (Of/)Go 00,700 =(0t/)^oo 

12. (Of/oo) oo^oc =(0t/^oo) + 

13. GO/GO oo^oo = -oo/yoo + oo 

When the relative and non-relative operations occur 
together, the rules of the calculus become pretty com 
plicated. In these cases, as well as in such as involve 
plural relations (subsisting between three or more ob 
jects), it is often advantageous to recur to the numerical 
coefficients mentioned on page 187. Any proposition 
whatever is equivalent to saying that some complexus of 
aggregates l and products of such numerical coefficients 
is greater than zero. Thus, 

^A>o 

means that something is a lover of something ; and 

JW<, > o 

means that everything is a lover of something. We 

1 The sums of page 188. 



THE LOGIC OF RELATIVES. 201 

shall, however, naturally omit, in writing the inequali 
ties, the > which terminates them all ; and the above 
two propositions will appear as 

The following are other examples : 



means that everything is at once a lover and a benefac 
tor of something. 

means that everything is a lover of a benefactor of itself. 



means that there is something which stands to some 
thing in the relation of loving everything except bene 
factors of it. 

Let a denote the triple relative " accuser to of ," 
and the triple relative " excuser to of . Then, 



means that an individual i can be found, such, that tak 
ing any individual whatever, j, it will always be possible 
so to select a third individual, k, that i is an accuser to 
j of &, and j an excuser to k of i. 

Let TT denote " preferrer to of ." Then, 



means that, having taken any individual i whatever, it 
is always possible so to select two, j and k, that i is an 
accuser to j of &, and also is either excused by j to & oi 
ls something to which/ is preferred by k. 

When we have a number of premises expressed in this 
manner, the conclusion is readily deduced by the use of 
the following simple rules. In the first place, we have 






202 THE LOGIC OF EELATIYES. 

In the second place, we have the formulae 



In the third place, since the numerical coefficients are 
all either zero or unity, the Boolian calculus is applicable 
to them. 

The following is one of the simplest possible examples. 
Required to eliminate servant from these two premises : 

First premise. There is somebody who accuses every 
body to everybody, unless the unaccused is loved by 
some person that is servant of all to whom he is not ac 
cused. 

Second premise. There are two persons, the first of 
whom excuses everybody to everybody, unless the un- 
excused be benefited by, without the person to whom he 
is unexcused being a servant of, the second. 

These premises may be written thus : 



The second yields the immediate inference, 



Combining this with the first, we have 

2 x 2 u 2 y 2 v (e uyx + s yv b vx ) (a xuv + s yv l yu }. 

Finally, applying the Boolian calculus, we deduce the 
desired conclusion 



U yxaxuv + fyJyu + xuA-:r)- 

The interpretation of this is that either there is some 
body excused by a person to whom he accuses somebody, 
or somebody excuses somebody to his (the excuser s) 
lover, or somebody accuses his own benefactor. 



THE LOGIC O* RELATIVES. 203 

The procedure may often be abbreviated by the use 
of operations intermediate between II and . Thus, 
we may use H r , II", etc. to mean the products for all 
individuals except one, except two, etc. * Thus, 

n/n/%+^ 

will mean that every person except one is a lover of 
everybody except its benefactors, and at most two non- 
benefactors. In the same manner, S 7 , ", etc. will de 
note the sums of all products of two, of all products of 
three, etc. Thus, 

(W 

will mean that there are at least three things in the 
universe that are lovers of themselves. It is plain that 
if m < n, we have 

U m - IP 2 n - ^ m . 



(n/V) (n/%) -< np+ fai . yi) 

Mr. Schlotel has written to the London Mathematical Society, 
accusing me of having, in my Algebra of Logic, plagiarized from his 
writings. He has also written to me to inform me that he has read 
that Memoir with " heitere Ironie," and that Professor Drobisch, the 
Berlin Academy, and I constitute a " lederliche Kleeblatt," with 
many other things of the same sort. Up to the time of publishing 
my Memoir, I had never seen any of Mr. Schlotel s writings ; I have 
since procured his Logik, and he has been so obliging as to send me 
two cuttings from his papers, thinking, apparently, that I might be 
curious to see the passages that I had appropriated. But having ex 
amined these productions, I find no thought in them that I ever did, 
or ever should be likely to put forth as my own. 



TIIE END. 



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