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 Philodemus1 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. Sensation4 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 gods6 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 logic3 consists in in ference from the observed to the unobserved. This was called a sien-iiiference. According to Epicurus there are two methods4 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 thunder5 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 tions1 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 ?;, Ka06, 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 2n~1 inches in di ameter. We should secure, as in the syl logism, 2n — 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.

(5r) 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 0 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 7.25 + 26 + 25, and that included by the second member is 25 + 3.26 + 7.26 + 28, 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 al9 «2> az . . . and con notes the qualities al9 «2, «3 . . . , then

a = al a2 + . . . 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

a1 + a2+... + ai-< bL + b2+ bs+ .. . 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 0 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 0 = "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 0 = " x exists, and at the same time does not exist," — a propo sition which is false whatever x may be.

Since the symbol 0 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 pl p2 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 0 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) is (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 0 + ft V) C J

(» v) + 0 v) v 0 + ^ 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 —

Ia. 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 —

Ib. 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 ;

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

Id. 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 + a£x 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 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 0 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 — 0 (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 = 0 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 = 0 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- O9)

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

W9 — 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 xm, 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 2n. 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 22W. 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, 0 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
0	0	0	0	0
0	0	0	1	1
0	0	1	0	2
0	0	1	1	3
0	1	0	0	4
0	1	0	1	5
0	1	1	0	6
0	1	1	1	7
1	0	0	0	8
1	0	0	1	9
1	0	1	0	10
1	0	1	1	11
1	1	0	0	12
1	1	0	1	13
1	1	1	0	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 216 or 216 — 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 23, the number of possible universes is 223, = 256, and the num ber of possible propositions with all degrees of ambiguity is 2256. 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.

0

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.	. .	0	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, pl9 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, pl and p2 are not true together, and pz 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., ply p^ ps 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 pip2pz v> ig °^ tne tyPe 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)p3 v> — belongs to type VI. ; and the identity,

(a \7 b) (c V d) — (ab +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 F19 Some CT is.*; « « Fu.

These two forms are so related that

= oo

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

one equation

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

FU + FU= oo;

that is, Fu and Fu 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 Fl is not jFi, but (^), which, according to the above, = Fu.

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), ............ (a5)

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

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

(ab + afyi ............ («5 + ab)

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

(ab + al\ ............ yn + afy

(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 + 5s),	. (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 2.223,= 512, different propositions. With n terms the total number is 2.22".

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 J7" implies " some U is I7."

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

U is F;" that is, F0 = F19 and since a proposition whose suffix is 0 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 Fu 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 J7, 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 Fv. Sometimes Fe will be written as a form inclusive of both the forms F: and Fu ; 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 Product 1 and the conclusion is equality ;

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

(1) F& = (FG)

(2) F,Gu-<

(3) FUGU -< oo.

FU+GU= (F+G)U, (10 Fu+G,-< (F+G)U, (20 F1+G1-<(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, Fu Gu —<^ oo. If it be known that u and u' have any common part, then for this common part FUG-U> -< (FG)UU,. Thus if u = f Z7, and u'=%U, then Fu G-u, -< (FQ-\,, , where u" = uu'=^U. Since we evidently have (FG-)U^^ FUGU, 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 Fl 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 Efy A') ; and any proposition Gu of the second column is equiva lent (I/) to the sum of one or more of the propositions 7, (9, J', 0', the two accented letters representing the particular propositions added by De Morgan. Thus Fl = U«, and II Gu = 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-< 0 + 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 jPe, where e may be either 1 or u. If I7 be a polynomial of the class terms, a,b,c,... x, y, z, then x, y, z may be eliminated from Fe by erasure, provided no aggregant term is thereby destroyed. That is,

Ft -< F,',

where F1 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." Fe 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. F1 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 Ft' 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

Fe -< (m = mF)., or Fe~< (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, «0 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 + F1) , 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 Fl = 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 A1. 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 F1 ,

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 Fl 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 Fl and Fu the polynomial F has been regarded as a function of class terms a, £, etc. Suppose a, b, etc. to be prepositional terms like Fl and Fu, 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 F19 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^ Fu 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 I7 be a polynomial function of class terms, a, 6, etc. Then let us consider the following system of six propositions :

Fuv, meaning "some part of U, during some part of V, is F" Ful) " " some part of U, during every part of V, is F" Flv, " "every part of U, during some part of V, is F? Fufl, " " 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" Fn, " "every part of U, during every part of F, is F."

By thus introducing the element of time, three varieties of the proposition Fu are distinguished, — Fuv, Ful, FU>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 F1 are also distinguished, — Fn, Flv, Flv,. 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,FlvFulFuv) and F^ + and since same is included under some, we have

Flv,-< Fln and F^ -< Ful. The following pairs of propositions,

Fuv and Fu ,Ful and F* , FU>1 and Fl9, 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, Fui, FIV, Fuv form a system of propositions by themselves, but it is seen that FW and Fu>i must be added to the system, in order to contradict Fu} and Flv. Mr. Peirce pointed out to me that these propositions are really triple relatives, and are therefore six in number. Fn, 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 Fn -<^ Fw

we get Fvv + Fuv = GO,

and ^ii^ii = 0;

hence Fn and "Fn are " contraries " of each other, and Fuv , Fuv are " sub-contraries." In the same way F& -< Flv

gives Fvl + Flv = GO,

and Flv,Fv>! = 0 ;

that is, Fw and F^ are contraries, and Fvl, Flv are sub- contraries. The line over F affects only F, not the suffices. Thus the negative of Fn would be written (3y, not Fn.

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

so F10 — Fu ,

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. Faji 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

FulGn-<(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

Fuv + GUV=(F + G)m,

Flv+ Guv-< (F+ G)uv,

etc.

But, by an exception to the

rule, do not have FulGlv-< (FG\V since Glv is not of the

form (Gj)v.

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

G)

and

(Fu)v + (Gu)v = (Fu + Gu)v =((F+ So in general we have

and

(Fa+

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 + U1 = 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 Fa^ 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)afi -< (a + b + cd + e)afi,

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)ap -< (c -< a + b + and, in general,

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

then we have

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

92 ON A NEW ALGEBRA OF LOGIC.

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

or H (S^u + 3Gul + SJGT^ + 2Klv + 2Llv + Muv),

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 Pu + 2 Q^ or PJIQU (see page 79) ; so that $1? <&v will be of the forms

(Pu+2Qi)i, WI ft,)..

It is clear that (PM + SQ^V = Puv + 5Qlv, and that (^PinQu)l = PnnQul; but for the two forms of $M <PV 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,

(Pu + 2Qv)r, (P^nft,),, (pv

The negative of <fry is $r, which is of the form ¥*„. So the negative of Xv is Xu, which is of the form flu. 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 flu 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", V9 W, proceeding just as before we should find that the number of fundamental propositions with three suffices,

•^in > ^uio F«a.> Fuvw, 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, Fl and Fu, are dual relatives. Ft means "F is a description of every part of U-, " and Fu means "F is a description of some part of £7." Thus Fl and Fu correspond to the two funda mental dual relatives. So in § 2, Fn is a triple relative term, meaning 11F 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 0 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 0 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-<^ 0 is equivalent to (XY)0, 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 jP0 and Fq are the two fundamental forms of proposition in this method, arid the rules of inference by combination are

F0G0=(F+G\ F,Gq-<(F+G)<

F+ G =

Fq+ GQ-< (FG)q F0 + G0 -< (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 0 as follows : —

XY V, equivalent to XY -< 0,

XY\[, " " XY^Q.

Thus F0 and Fu 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

FQGU-<(FG)U

FuGu-< oo.

Elimination from F0 is performed by multiplying co efficients ; from jPM, by adding them.

One more method remains to be noticed, — the negative of Miss Ladd's method, in which A and 0 are expressed

as

o>-< X+ Y,

and where F1 and Fq are thus the two fundamental forms of proposition. The rules of inference ly combination are

Fq+Qq=(FGT)q

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

oo

and elimination from Fl is performed by addition of co efficients ; from Fq, 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', 0', 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', 0', A have already been represented (see page 76) respectively by

OK A NEW ALGEBRA OF LOGIC. 99

and also, since Fl — FQ, by

(ab)u, («5)o, (a5)«, (a&)0, 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)-lx (sm)-< (sp)-,

but from A and 0 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 : —

-lm^) (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 2.2n 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-lm 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

(r1*)-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, 0 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	0	sp	0	0	0	Sp	sp
sP	sP	sp	sp	0	0	0	0	sp	sP
sp	0	sp	0	0	o	0	0	sp	0
sp	sp	0	0	0	0	0	0	o	sp
SP	0	0	0	0	0	0	0	0	o
Sp 7P	0	0	0	o	0	o	o	0	0
o	0	o	0	0	0	o	0	0
sp	sP	sp	sp	0	0	o	o	0	sp
w	sp	Sp	0	sp	0	0	0	sp	0

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 p1 and not in the relation p". Again,

p' -f p" = number of pairs in circle, or square, or both + num ber in both = (p1 -I- p") + p1, p",

in which— according to Mr. Jevons's notation — (p1 -|-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 pr. 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

0'] + [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 : —

Pn-2]) Pre-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 prp" 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 X2°°.

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 p1 must hold ; and no relative sequence of the form p'p" or p'pn 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 pn or p

00 and

or

I

P"

P'P" _ P'P"

That is, the average number of sequences pfpn per each p" is the same as the average number of sequences p'pn 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 pn, 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 ..... |. pM] 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'. . . pn. In the case of incompatible relations we have the equation

[p'-lV'-l ..... |-PB]

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] — 0 ; 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...\m9n]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- • -mT

= (M P] M - - -