alexander mcfarlane principles of the algerbra of logic

3analyticalmethodof reasoning about Qualityisan Algebra, which coincides with the Algebra of Quantity when the symbols are integral, but is a generalised form of the work is taken up wit

PRI NCI PLES

OF THE

WITH EXAMPLES

ALEXANDER MACFARLANE

READ BEFORE THE ROYAL SOCIETY OF EDINBURGH

EDINBURGH : DAVID DOUGLAS

DAVID DOUGLAS.

LONDON HAMILTON, ADAMS,ANDCO.

GLASGOW JAMES MACLEHOSE.

PWlleS

631004

OF THE

valueoflogic, thevalueof mathematics, the valueof logic inmathematics,

Shall we then err in regarding that as the true, scienceof Logic,

entirechain ofitssecondary consequences,and furnishes for its practical applicationmethodsof perfect generality. Letit be considered whether

inanyscience,viewedeither asa systemof truthoras thefoundation of a

practical art, therecan properly beanyothertestof thecompletenessand

establish. BOOLE,Lawsof Thought,p 5.

It is curious to compare the properties of these quaternion sym

grandscienceofmathematicalanalysis,byprocessesremarkablysimilar to

eachother, reveals to us truths in the scienceofposition farbeyond the

powersof thegeometer,andtruthsofdeductive reasoningtowhichunaided

p 50.

THESE Principles were originally contributed to

by the Secretary Qth October 1878, and in a supple

honour of reading an Abstract before the Society at

the meetings of i6th December and 2Oth January.

In the interval between the 5th November and the

present time I have improved several of the demon

strations, introduced illustrative matter, and prepared

the collection of examples The work, in its present

state, forms an elementary treatiseon the science of

I considerit proper to state that the theory of the

operation of the mind in reasoning about Quality,

which is advanced in this work, occurred to me five

years ago; and that I have directed towards its

development the whole of my subsequent study of

the Mathematical, Physical, and Natural Sciences,

which are embraced in the curriculum forthe degree

of Doctor of Science (Mathematics) at the University

of Edinburgh.

ALEXANDER MACFARLANE.

EDINBURGH, i$d January 1879

REV PHILIP KELLAND, M.A., F.R.S.

PROFESSOROFMATHEMATICS INEDINBURGHUNIVERSITY

of tfje iftogal Socfetg of OEoinfourgfj

AS A MARK OF RESPECT

BY

A FORMER PUPIL.

PAGE

I, TheScienceofFormalLogic anAlgebra, i

II. Universe andCharacter, 5 III. Thesign=, .15

IV Thesigns+and - , 17

V Thesignsxand-f- , 20

VI Ruleof Signs, 25

VII IntegralSymbols, 26

VIII OntheEquationas expressing a general proposition, 28 IX The principle of Identity and the Axioms of Im mediateInference, 36

X AxiomsofMediateInference, . 40

XI ConditionsforaCharacter beingSingle, . 42

XII Thesigns ofinequivalence > and < , 53 XIII Division, 54

XIV Expansionof a function of anumberof independent symbolsinterms of the primaryparts intowhich the universeisdividedbythe symbols, . 61

XV. Definition, . 69

XVI Inference from one or more equations of the form x=m (Categorical), , 7 XVII Inference from oneor more equations of the form xy=m (Hypothetical), 81 XVIII Oncertainformsof the disjunctive equation, . -106 XIX. TheAristotelianformsofinference, .113

XXI. Fundamental relations between the single functions

of anumberofindependentcharacters, . 122

XXII GeneralMethodofdeducinga conclusion of a required

XXIII OnBoolesGeneral Method, . 131

Examples, 135

ALGEBRA OF LOGIC.

I. THE SCIENCE OF FORMAL LOGIC AN

ALGEBRA.

the matter, that the theory of Necessityand the theoryof

Probability are the complementaryparts of one whole, it

is nevertheless true that the foundations of the general

science, of which theyform the parts, were not laid until

quite recent times. The merit of conceiving and under

taking this important unification is due in some measure

to De Morgan,but principally to Boole

2. That the science of inference is capable of being

treated analytically, may beinferred fromthefact that the

ordinaryrules about Conversion and Syllogism are estab

lished bya comparison of circles, taken to represent the

terms of thepropositions considered. In one of the best

modern manuals of Logic, it is stated that the testing

whethera given combination of premises leads to avalid

inference, and the proof ofthe validity or invalidity,must

depend on the comparison of the spheres, within which,

according tothe premises, the notionsunderconsideration

find application; andthatthese spheresare made apparent

tothe senses bygeometrical figures (especially bycircles)

whose reciprocal relations agree with the* relations of the

spheres ofthenotions toeachotherin allrelations essential

for demonstration

(UeberwegsLogic^ translated by Pro

fessor Lindsay, p 379.) The introduction of these dia

gramsis commonly attributed to Euler

2 The of

3. Corresponding to this graphical method, which con

sists intheuse of diagrams, there isananalytical method, which consists intheuse of symbols Therelative advan

tagesand disadvantages ofthe two,whenappliedto Quality,are preciselythe same aswhenapplied to Quantity The

diagram exhibits an individual caseof the given data with

all the clearness of the concrete; on theother hand, theanalytical expression separates theessential relations from

theaccidental, withwhich they must be mixed up in any

individual example

4. The reason why the operations of Booles calculus

appear mysteriousandits employment difficult, is, that thecalculus is not founded upon a sufficient theory of theoperation ofthemindinreasoning about Quality That it

is not all that a Logical organon ought to be, is evident

from what Venn says in Mind, vol i p. 484: The dis

tinctive characteristic of Booles system is the boldness,

notto say audacity, with which he carrieson his processesthrough stageswhich have no logical or othersignificationwhatever, that is, which admitofnopossible interpretation

provided only theyterminate in an interpretable result

Boolehimself claims nothing higher for hiscalculus Hewould, however, have objected to the statement which

Professor Jevons makes (Principles ofScience, p 71), that

Boole importedtheconditions of numberinto thescience

of Logic, and produced a system which, thoughwonderful

5. It is the object of this littleworkto investigate thefoundations of the analytical method of reasoning about

Quality, with special referenceto the principles laid down

byBooleasthe basis of his calculus,andto theobservations

which have been published by various philosophers con

cerningthese principles I bring forward anew theory of

the operation ofthemindinreasoning aboutQuality,which

enables me to correct Booles principles, and place them

on a clear rational basis I endeavour to show that the

Formal Logic an Algebra. 3

analyticalmethodof reasoning about Qualityisan Algebra,

which coincides with the Algebra of Quantity when the

symbols are integral, but is a generalised form of the

work is taken up with the investigation of problems by

means ofthisalgebraic organon, especiallysuch problems

as are suggestedbytheordinary Logic.

6. Logic, as theAlgebra of Quality,is aformalscience. It

investigates thegeneralproperties ofthesymbol ofQuality,

and by means of these properties deduces equationswhich

are true generally, or combines such equations with data

of given forms It is not its province to considerhow a

particular formofdatum caninanycase beasserted to be

true that subjectofinvestigationbeinglefttotheTranscend

ental -Logic; it is sufficientthat examples of such a form

occur, ormay occur,in the practical or theoreticalactivities

ofmankind.

7. The properties of the symbol of Qualityarenotlaws

of thought in the common acceptation of that term For

the properties of the symbol of Quantity, on which the

ordinary algebra is founded, are held not to be laws of

thought, but to refer to the actual constitution of things

j

and there is no difference in the two methods, when

developed, which indicates the existence of such a dis

tive, it is so onlyinthesame sense inwhich the basis of

the scienceof Quantityissubjective Thereisground for

believing that the true reason whythe former science has

remained so stationary is, that there has been too much

introspection into the individual mind in the hope of

rinding lawsof thought there, and too littlecontemplation

of the form and nature of the truths of Science The

logician assumes that all men reason equally well about

Quality, fallacies beingpossibleonlybyamomentarylapse

ofattention; butthe mathematician neverassumes thatall

men reasonequally wellabout Quantity.

4 The of

8. Boole entitled his great work on reasoning An

Investigation of the Laws of Thought, on which arefounded the Mathematical Theories of Logic and Prob

abilities, and in several places he says that the Laws in

question are subjective in a sense in which the Laws ofQuantityare not. He considers

in particular to be a subjective law; but I have endea

voured to show (Art. 118) that it is a special condition,

which the symbol of this Algebra mustsatisfy inorder to

be of aparticularkind

9. No one, I suppose, contendsthat the propertiesofthe

Chemical Symbol, or ofthe Quaternionic Symbols,arelaws

of thought. Since the corresponding properties of

^

the

surely betterin every case to consider the actual constitu

tionof things assuggesting rules for thought to the mind,

ratherthanthemind imposinglaws of thoughtuponitself.

10 Logic, as the Algebra of Quality, is atrue organon

It can determinewhether a conclusion of a required form

can be deduced from dataof given forms; and ifso, what

that conclusion is. Itcan manipulate complex data, as is

shown in* the examples appended Bacons judgment

Syllogismus ad principia scientiarum non adhibetur, ad

media axiomata frustra adhibetur, quum sit subtihtati

naturae longe impar however true of the scholastic ex

position ofthe syllogism, does not applyto theAlgebraof

Quality; forthe latter can be madetodiscoverprinciples,

andto imitate to some extent the subtlety of Nature Itmay be said (to adapt a remark of De Moivre) that in

numerable questions in the theory of necessary and pro

bable reasoning can be solved without any manner

^

of

trouble totheimagination, bythemereforceofthenotationsuppliedby thisAlgebra

The Algebra of Quantity is acknowledged to be the

weapon for the philosopherwho attacks the Experimental

Formal Logic an Algebra. 5

Sciences; the Algebra of Quality is the weapon for the

philosopherwhoattacksthe Sciences of Observation

11 Thusviewed, Formal Logicis not the shortand dry

studied Booles Calculus, maywell imagine that thetheory

of reasoningwas not completed byAristotle; and that so

far from any System of Logic having ever beenwritten,

there is stillneed toconsider thefoundations

II UNIVERSE AND CHARACTER.

12 Boole in his analysis of language draws nodistinc

tion between Substantive and Adjective; he considers their

function in reasoning to be the same He says (Lawsof

Thought, p 27), Thesubstantive properand theadjective

may indeed be regarded as differing only in this respect,

that the former expresses the substantive existence of the

individual thing or things towhich itrefers; the latterim

universally understood subject "being" or "

thing/ itbe

comes virtuallya substantive, and mayforall the essential

purposes of reasoning be replaced by the substantive

Accordingly, heusesthesymbol xtodenote men or good

things or white things or horned things, as the case

may be For instance: he says, if x alone stands for

white things and y for sheep, let xy stand for white

sheep; and in likemanner ifzstands for horned things

and x and yretain their previous interpretations, let zxy

represent hornedwhite sheep.

13 Again; when investigating the operations of the

mind, which are implied in the use of language asan in

strument of reasoning, he finds no difference inthe

opera-6 Universe and Character.

tion expressed by asubstantive fromthat expressed by an

adjective He says that there is a universe of discourse;but this universe is not one described by a substantive

In every discourse, he says, whether of the mind con

versing with its own thoughts, or of the individual inhisintercourse with others, there is an assumed orexpressed

words we useareunderstood inthewidestpossible applica

with those of the universe itself. But more usually we

confine ourselves to a less spacious field. Sometimes, in

discoursing ofmen, we imply (without expressing the limita

conditionsthat we speak, as of civilised men, orof men in

the vigour oflife, orof men under some othercondition or

withinwhich all the objectsofour discourse arefound,that

Lawsof Thought, p. 42

14 Fromthepassage justquoted, as well as from many

others, itappearsthatwhat Boole means bythe universe ofdiscourse is not the objects denoted bya Universal Sub

a limited portion of the physical universe, with all the

mental or physical, ponderable or imponderable, atomicor

complex

15 The substantive men expresses an operation of

election from the universe of all the beings to which the

term men is applicable; the adjective good in com

bination, as good men, directs us still further to elect

mentallyfrom the class of men all those who possess the

further quality of good; and if another adjective werepre

fixedto thecombination, itwoulddirecta similaroperation

upon good men. In short, he supposes that the mind

Universe and Character. 7

always proceeds alongthepredlcamental line ; whereas that

is onlyone mode ofitsprocedure

16 In consequence of this analysis, the subjects of

thoughtin Logicandin Arithmeticare said to beperfectly

distinct

; and it is not of any importance to compare the

symbols of logic with the symbols of quantity generally

Attention isdirected so exclusively to an Algebrainwhich

the symbols xt y, z, etc., admit indifferently of thevalues

o and i, andof these alone, thatsome logicians have sup

posedthat thesymbols canhave no other value.

17 Another consequence of this -analysis is, that Boole

is obligedtomake anew and independent investigation of

Secondary Propositions In the case of Secondary Pro

positions, the proper interpretationof the symbol i isheld

gested, whetherinthe case ofPrimary Propositions i does

sign of identity = connecting the members of the corre

sponding equation, implies that the thingswhichtheyrepre

sent are identical, not simply that they are found in the

same portion of space. The reasonwhy the symbol i in

SecondaryPropositionsrepresentsnot the universe ofevents,

but the eternity in whose successive moments and periods

theyare evolved, is,thatthe same signofidentityconnect

ing thelogicalmembers ofthecorresponding equationsim

same Lawsof Thought, p 176

18 The principles of the Calculus of Identity become

takinginto considerationthedifferenceofthefunctionsofthe

Substantive, andofthe Adjective used inanattributive sense

The objects expressed by the common noun, or rather

universal term of a proposition, constitute the universe

of the proposition the actual whole considered by the

mindinformingthe judgment The attributive adjectives,

8 Universe and Character.

whether oneormore,whichappear inthe proposition,refer

tothat subject, andnotto things in general. In thinking

of sheep that are white and horned, I do not considerwhite things or *

hornedthings. It is even questionablewhetherthemind can considersomeadjectives as denoting

classesofthings. Can we consider smallthings or wise

things or primarythings ? Booleremarks, with reference

to this very attribute wise, that, before denoting it by asymbol, we must consider whether it is to be used inits

absolute sense or only relatively. But small has no

absolute sense Nothing byitself/Aristotle lays downin

is described as great or small. A moun

little,"and a millet

seed "large," from the fact of the one being greater, and

the other less, in respect ofthings of thesame nature It

isthisreference to thingsof thesame naturethatI wish to

drawattention to

19 As quantities have a certain abstract meaning in

themselves, but no definite meaning unless withreference

to agivenunit; so qualitieshaveacertain abstractmeaning

inthemselves, but no definitemeaningunlesswhen referred

to a givenuniverse

20 Let the particular kind or collection of objectsconsidered in any judgment or series of judgments be denoted byacapitalletter U a symbol usedin an analo

goussenseby-DeMorgan Whenthesame kind or collection of objects is the subject of all the judgments con

sidered, U need notbe expressed,butisto beunderstood

Let an attribute, character, or quality, be denoted by asmallletter, as x

21 This modification of Booles notation brings out the

contrastbetweenthe SubstantiveandtheAdjective; which

is indeed only one form of the general contrast between

thatwhich isthe subjectof the operations of thought and

the operations themselves. Another common form of thecontrast is that, made prominent in the Theory of Prob-

Universe and Character 9

ability,between the event and the wayinwhich itcan

happen

If Udenotethe Membersofthe HouseofCommons at

thepresent time/x maydenote c

Liberal or Conservative

Or if Udenote triangles/ x may denote isosceles or

equilateral

22 Boole, in hisinvestigation,generally considers acom

binationof a Substantive withanAdjectiveprefixed. Some

languages, however, show, by a difference of position

before oraftertheSubstantive that the Adjective may be

usedintwo senses, viz., as formingpart of the Substantive

merely, or as equivalent to a relative phrase In the

English language, wherethe adjective is commonly placed

before the substantive, the distinction referred tois brought

outbyemphasis Theprefixed adjective,whenemphasised,

is equivalentto arelativephrase; when not emphasised, it

ispartofthe subject ofthought.

23 For example: Ifind on lookingupa Polyglot Bible

that the proposition of Proverbs xv 20 is expressed as

follows:

Awise son maketha gladfather

Ytoscro^osev(/>paiveiTrarepa.

Filiussapiens laetificatpatrem

Ein weiserSohn erfreuetdenVater

Lenfant sagerejouit sonpere.

Elhijo sabio alegraalpadre

Here the subject of discourse is Sons ; and it is to be

observed that while thetwo Teutonic languages placethe

conditioning attribute before the subject, the others putit

after.

24 Boole holdsthat PrimaryPropositionsreferto things,

and Secondary to facts; and that the idea of time is in

volved in the Secondary. Now there are propositions

relatingto facts which donotinvolve time, or a collection

ofportions of time, as theunderlyingsubject; forexample,

io Universe and Character.

those which refer to place or a collectionof places. We

have not onlythe relative *

when, butthe relatives where and who Henceiffact-propositions, whichrelate tothe

identity ofportions oftime, required aspecial investigation,those whichrelateto the identityof portionsofspacewould

also require a special investigation. If, however, we draw

a contrast between the subject and its characters, one

investigation suffices for all the different kinds ofsubject.Instead of two I s, of which the one means the actuallyexistent universe, and the other eternity,there isaninfinite

number of 7s,any one of which may be the subject of

discursive thought.

25 This view of anessential difference in thefunctions

of the Common Noun and Adjective is supported by the

(Lecturesonthe Scienceof Language,vol i.

distinguished from the verb byits collocation

in the sentence In the Aryan languages no predicative

rootcan by itselfformaword; in ordertohave a substan

tiveit isnecessarytoadd a demonstrative root, thisforming

thegeneral subject of whichthe meaning contained inthe

root is tobe predicated. If Boolesview ofthe operation

of the mind were correct, we should have onlypredicative

roots

a strong contrast between Substance (ovo-ta) and Quality

(TTCHO V) ; and between Primary and Secondary Substances

By Substance is meant a particular thing (root rt); by

Quality that which is in a subject. In the case of the

PrimarySubstance, the thingsignifiedisindividual and one

in number;in the caseof the Secondary, thethingsignified

Universe and Character.

involves aquality, but so as todenote aparticular kind of

substance It is the characteristicofSubstancethat being

one and the same in number it can receive contraries;

while it is the characteristic of Quality to be that with

respectto whichthings are said to belike or unlike

27 ThePrimitivejudgmentis so called because it does

universeas onesubstance having all the physical changes

which occur for accidents For example, the judgment

It rains

andoftheseequates the present with raining.

28 Theordinary Eulerian Diagrams donot represent the

wholeofthought,but leaveitindefinite

; unless wesuppose

ittobe represented bythefinite sheet, on which the attri

butes are representedby circles

29 Uis in general made upofa typeand certainfinite

of the Noun, of which Professor Max Miiller speaks, and

the limitations to the demonstrative part. The most

in this aspect, may be looked upon as logical variables

We maysuppose the Timeto beconstant, andconsiderall

the Usthroughout a given region; orwe maysuppose the

individual tobe constant, and consideritssuccessive states

within a given portion of time The zoologist, when he

compares themembers of a genus, takesthemin the adult

form, he follows anindividual throughits cycleofstates

30 It is with thenotion ofthetypethat questionsabout

Essence and Abstract Ideas aremoreproperly concerned.

Berkeley draws adistinction between twokinds of abstract

mental separation attain abstract ideas of the more com

poundedbeingswhich includeseveral co-existentqualities.7

12 Universe and Character.

ProfessorFrasersSelectionsfromBerkeley,p 16 Hesays

also thatthere aretwo kinds ofabstraction tocorrespond.

Thedistinctionconsidered is correlative tothat between

Universe and Character

31 Arithmetical valueof U. Since 7signifies adefinite

collection of individuals of a given type, its arithmetical

valuemust be aninteger. The integeris ingeneralfihiral,

but may be singular or infinite. It is infinite when the

individual parts are notdiscrete but continuous Grammar

recognisestwo ofthesecases

It is interesting to consider howthe subject of thought

has naturally an integral value, while the operation of

thought has naturally afractional value, howthe relation

between the symbols is mirrored in the relation between

theirkinds ofquantity.

32 U also may be either real or imaginary. For

instance,thejudgment

Thegoat- stagis white

imaginary, it appears properto indicate that factby saying

thatthe arithmetical value of Uis o; whichistherefore,in

thisaspect, an integer.

33 TheUniverse holds thesamepositionintheAlgebra

of Qualitythat the Unit does in theAlgebraofQuantity

Itmay besaid tobe ageneralisedtmit

34 When Uis used to denote the subject of thought,

and x, y, etc., to denote operations on it, the symbols

meaning is supposed to be fixed throughout a discourse,

their arithmetical value must also be supposed fixed If

xdenotes asingle positive attribute, its value is a fraction

lyingbetweeno and i

; but if it isnegative, its value lies

between o and i. Suppose that we have a complex

character as xy; being compounded of two characters

x and y, whichare intheir statement independent of one

Universe and Character. 3

another It is then necessary to suppose that the arith

metical values of x and y are preserved independentlyof

the combination; for these symbols depend on Uonly

meaning of y is not independent of x; and y may have

several arithmeticalvalues according to the several orders

ithasin combination

35 Thusidenotes <

all or thewhole ; while o denotesnone i and oare to beconsideredasoperatingsymbols

of the same kind as x Some is an indefinite operating

symbol; butitgenerallycarries the additional meaning of

havinganarithmeticalvaluewhich isgreaterthannought

36 It is veryfrequently necessaryto express the arith

metical value of x A convenient notation is x Boole

37 The mind, when reasoning on matters such as are

discussedintheTheoryofProbability,considers aparticular

his Logic of Chance. The /s and ^s are in their first

lying between o and i. A dependent event involves

another event as a presupposition; and its arithmetical

value depends on that connection a circumstance which

alsoshows us that attributeswhichare independentin their

statement must be conceived as operating upon the uni

verse directly.

38 TheAlgebraof Qualityisthe more generalmethod.

It discusses the relations of the characters of a Universe,

whetherthatuniverse comprise one, several, or an infinite

number ofparts, and whetherthe characters change or are

independent oftime; whereas theTheory of Probability as

commonly stated (see Venns Logic of Chance, p 5),

supposes the universe to comprise a verylarge or infinite

number of individuals, and the proper arithmetical value

ofpto be acertain limitingratio, towhichthe actualvalue

of/ is continuallyapproaching the greater thenumber of

individuals inthe universe

4 Universe and Character.

39 There is an important distinction among adjectives

according as they involve beingorhaving Thecontrary of

theformeris formed byprefixing not, thatof the latter by

prefixing without. Let us consider a collection of balls

The attributes which, in the Theory of Probability, such

red, black These are attributes of being, since

they apply to the whole of the ball, or at least to the

whole of its surface But we may consider aball,which

butes as having a white stripe or having, a red stripe

are adjectives of having The adjective with a white

As the term character is the more general of the two, I

prefer touse itinstead ofquality.

40 The ordinary doctrine of

Subject and Predicate is

oftendeparted frominManuals of Logic without thewriter,

apparently,being aware of it. Forinstance, Ueberweg, by

thus leaving <

unchanged, and

manipulating only the characters regular and inscribable

41 De Morgan brought the idea of the Universe into

prominence Inhis Syllabus (section 122) he defines itas

meaningthewhole extentof matter of thought under con

sideration I donot considerthis definition as sufficiently

exact as

indicating precisely thatwhich intheAlgebraof

Quality corresponds totheUnitof theAlgebraofQuantity.

With him the term does not denote a definite subject of

The Sign = 15

operations, but rather limits within which elective, not

Ux, he denotes the resultbyX

42 By U{x=y}

is meant, that those Z7s which have the character x are

identical with those which have the character y; or, in

anotheraspect, thatwithin the universe Uthecharacterx

isequivalenttothe charactery.

It follows thattheequationistruearithmetically. If

*sajr,

the equivalence of two characters involves the equalityof

43 Thoughthe 7s,which havetherespective characters,

areasserted to beidenticalwithoneanother, the characters

expressed bythe term Equivalence. If the proposition is

but if it isverbal, theyderive their equivalence from con

vention Two characters may be equivalent in themselves,

or bydefinition,orby reality. For example: let U denote

triangles; x, equilateral;y, equiangular; then

U{*=y}

does not assert the identity, that is, theundistinguishable

samenessof the characters *

equilateral and equiangular,

but onlytheirlogical equivalence

44 We mayalso have an equation ofthe form

U{x = x}

where x denotes the arithmetical valueof x, and is sup

posed known. This equation also expresses the logical

16 The Sign =

equivalence of the members; but the second member is

indefinite asregards identity For example: let Udenote

then U{x =J}.

This is an example of a form of equationwhich bulks

largely in theTheory ofProbability.

45 Since x and y, when regarded as operating on a

definite universe,acquire an unambiguous and exactarith

meticalvalue, novalid objection can beraised againstex

pressing ajudgment of the kind considered asan equation.

Ithas, indeed, been rightlymaintained bythe opponents

ofthedoctrineof the Quantificationofthe Predicate, thatit

vanced in these Principles is, thatthebasisofthejudgment

aloneisdenotative; andthatthemembersofthejudgment,viz.,

the antecedent and theconsequent, are both attributive. The

contrastcomesout inthe factthat the arithmetical value ofthe subjectisintegral, while that of the antecedent andof

the consequentis generallyfractional

46 Itappearstobeof the essence ofanEquation,thatits

membersbeequalarithmetically This holds oftheChemical

Equation For example, in

Alcohol becomesWaterand Olefiant Gas

the arithmeticalvalue ofthe right-handmemberisequal to

the arithmetical value of the left-hand member. Also in

the Quaternion Equation

the arithmeticalvalue of a-fft is equalto that of

y.

The Signs + and

IV THE SIGNS + AND

-47 By U{x+y]

ismeant

7swhich are xtogether with 7swhicharejy/

x upon Uto the selectiveoperationy upon U Readwith

respect toasingle U, theexpressionmeans

A U whichis eitherxory.

48 Suppose that we have a collection of cubes, and

some haveadash Let the operation

xdenotethe separationof those having

a dot; and the operation y of those

having a dash Suppose x first per

formedj the collection is broken up

which have adot, and thosewhich are

without adot Thesubsequent opera

tionyconsistsoftwoparts,viz.,aselec

tionof those having adash, firstamong

those having adot,andsecondly,among

those without a dot; and therefore gives as aresult four

of thosehaving a dot to those having a dash, whetherthe

firstdivision, whichconsists of those having a dot and hav

inga dash,vanish or does notvanish

49 Since the sign is by definition the sign of the

operationwhichis the reverse of that denoted by +; and

since the sign -f-has its ordinary algebraic meaning, the

onlydifference

consistinginthefactthatthe symbols joined

involveidentity

; the sign must haveitsordinaryalgebraic

meaning, but modified by the above-mentioned circum

stance

I

18 The Signs + and

U{x-y]

means

Z/swhich arex minus Z7swhich are y

x and ydestroyoneanother, sofarasthey coincide; and

part

50 Consider thecollectionof cubes representedbyfig. i.

x y denotesthatthose having a dot are tobeseparated

cubeswhichare both x and

j>(division i)will be manipu

that division 2 ofthe figureistakenpositively, anddivision

3 negatively

51 If the operationsx and yare formallyindependent,

rs in the above example, the arithmetical value ofx+ycannotbeaffectedby any real relation existing between x

and y. It is simplythe sumof the arithmeticalvalues of

x andy. Theoperationy (fig. i),though subsequenttox,applies to the wholeof the universe; hence

=1+1in thecaserepresented.

The aboveisnot the viewcommonlytaken De Morgan

says (Syllabus, section 131), The moreclasses aggregated,

of the others, the greater the extension of the aggregate

term

52 If it were necessary for x and y in x+y to be

mutually exclusive, a restriction would be placed upon -f

which would provefataltothedevelopmentofan Algebraof

Quality But the imposing of the restriction would be a

mistake, for thebasis of our operations isnot aUnitbut aUniverse x-\-y does not necessarilydenote asingle char

acter; butwhat itin general denotesisequally intelligible,

asummationoftwocharacters

Similarly, it isnot necessaryforx inx ytobe inclusive

ofy, orfory tobe inclusiveofx In the Algebra of

Quan-The Signs + and 19

titythese conditionshold

; theydonot hold inthe Algebra

ofQuality,becausetheUniverseisageneralised Unit. The

Universe becomes a Unit when it contains onlyone indi

vidual

53 Definition ofSingle A symbol xis said tobe single

when it does not select any memberof the universe more

than once, and alwaysselectswith the samesign.

Observation Thisembraces exclusion,and bothpositive

and negativeinclusion

54 Ifwe have z=x-{-y,

andzisa single character, then x and y must be mutually

exclusive It is in thiswaythat the Algebra imposes the

z=xy,

55 Toexpressnotxin terms ofx

By not x is meantthe whole excepting x; hence it is

expressedby ix.

Thearithmeticalvalue of notx is i x In the example

ofArt 48, i xdenotes not having adot, andforms part

andformsparts 2 and 4.

56 If xispositiveandsingle; then ix ispositive and

single

For x being included in i, and i being single and

analytical proof, see Art 120

57 When x and y are intheirstatement independent of

one another, their combination by + and is subject to

theformal laws

(i)(2)

x-y=-y+x. (3)

58 The truth of the formal law (i) may be seen by a

consideration of the particular instance of thecubes with

2O The Signs + and

differentmarks If we first separate those having a dot,

then separatethose having a dash, and add the tworesults

together; we shall get thesame cubesin thefinal result as

when we commence byseparatingthose having a dash,thenseparate those having a dot, and add the two results

together

The formal law (2) showsthat independentcharacters,

which are not necessarily single, are subject to the same

law ofcombination as independent characters, which are

single

Theformal law (3) shows that as a symbol of opera

tion of the mindis quite as independentas -f

V THE SIGNS X AND -^

is meant

C/s which areboth x andy.

In common language bothJ

is frequently dropped Ac

cording to Boole + expresses the signification of *

and ;

while X properly expresses sequence It is, however, an

additionalargumentin favour of adifferenceof notation for

the Universe and its characters, and generally between

operations whichare subordinate and operations which areco-ordinate; that, when it is done, + expresses or and

X expresses and

60 Consider, as before, a collection of cubes, Art 48;

agreatnumber Asbefore,letxdenote

*

having a dot, and y having a dash

Let xseparateto theleft hand(fig. 2);

and then let yseparate to thebottom

The operation xy is a consequence,beingthe separation ofthe cubes,which

both have a dot and alsohave adash

I

FIG 2.

The Signs x and-r- 21

61 Whenthesymbols xandjyare independent that is,

has for any given U a definite arithmetical value This

value, however, is notdeterminablefromthose ofx and y

but theygive limits tothe value. Considertheexampleof

the cubes (fig. 2). It isevident that xycannotbegreater

(include more) than x, or greater than y; and that it

cannot be less than (must include)x-\-yi, oro Hence

its arithmetical value cannot be greater than x or y,and

cannot beless thanx+y i,oro.

62 It is important to draw a distinction between the

independenceoftheoperations x andy, and therelation of

these to one another as characters The operation y is

independent of x, when it isnot confined byitsexpression to

Ux, butapplies to U In the Theoryof Probabilityit is

assumed that,when x and y areindependent,

xy = x Xy.

Butthisisan assumptionresting onthe supposition that the

assumedthat thetwostraight lines, whichtogetherseparate

theyinfig. 2, form onestraightline. It is, no doubt, the

bestassumptiontomake, when we donotknow howthetwo

portions of the linelie; and is of the same natureas the

doctrine that the valuesoftwo charactersnot beingknown,

theyare to be assumed to be equal. We have supposed

that the Universe contains a great number of cubes, in

order thateach portion oftheyline may be supposedtobe

straight

63 Venn shows the truenature oftheassumption in his

Logic of Chance, p 157 : For the establishment ofthe

rule under discussion, it is both necessary and sufficient

that the division into classes caused byeach of the above

bythe otherdistinctioninthe same ratio inwhich it sub

divides thewhole Iftheindependence begrantedandso

22 The Signs x and

definedas to mean this, the rule of course will stand, but,

without especial attentionbeing drawntothe point, itdoesnot seem that theword would naturallybesounderstood

Thedistinction maybecalled thatbetween formal andrealindependence

64 But one character may in its statement involveanother character, so as to be formallydependent on the

latter. Let U denote the universe of objects, a any

character, and x a character which is formally dependent

Uswhichare a andof these suchas are x;

or 7swhich have a which have x

The symbol xoperates upon Ua, notupon U

The arithmetical value ofx is measuredwithreference

to Ua, not to U Hence

ceed in the predicamental line. Boole supposes that it

alwaysproceedsin thatmanner. In Uxy he considersjy to

operate on Ux, in fact, tomean Uxy, and supposesthatit

canpreserve the same signification in Uy. But in fact its

meaning and arithmeticalvalue may both bechanged.

66 In the ordinary Algebra \ \ means one-half of a

third or one-thirdofahalf. The one operation is formally

dependent on the other. But we haveshown,Art 61, that

intheAlgebra of Quality

than the Algebra of Quantity

67 Suppose thatweconsiderthe Mammalia.

Then an example ofax is

having redblood corpuscles

whichare oval;

The Signs x and

and an example ofa^is

having redblood corpuscles

whicharecircularanddiscoid.

Itisthefunction ofthe RelativePronountodenote depend

69 Since ax is equivalent to an independent character

of arithmetical value ax, the laws that are true of inde

pendentcharacters applytoaxasa whole

70 Also, the same laws apply to characters which are

co-ordinately dependent upon a common character For

Observation It is supposedthat x is such that it can

befreed ofits dependence on a;whichisnotpossiblewhen

72 Deduction ofthe meaningof-rfromthatof

X-Letxy=ixz,

then 4- being the reverse of X is to

have such asignificationthat

xz

It is evident fromfig. 3, which repre

sentsa case of

FIG 3.

24 The Signs x and-^

that is indefinite Hence the proper reading of

althoughx and y bekeptconstant

COR. is notingeneral= i

73 Theconditionyzisalwayssatisfiedbythe fractions

of ordinary Algebra; butit isnotsatisfiedbythe fractionsof

theAlgebraofQuality, unless asa

special case Thispecu

liarity

f^-notnecessarilyequal to i\ was clearlypointed

out byBoole Itis commonlysupposed, and I thinkthat

Boole himself supposed,that it istheonlypeculiarityofthe

symbolsofthisAlgebra Forexample, ProfessorRobertson Smithsays: Thereis onelimitationonlyto our right to

The Signs x and 25manipulatelogicaland mathematicalidentitiesby thesame

tionof Ueberweg, p 569 Now the fact is that each sign

has a generalisedmeaning.

74 Boole interprets to mean an indefinite class

o

symbol; but this interpretation, he says, cannot except

upon the ground of analogy be deduced from its arith

metical properties, butmust beestablished experimentally

Lawsof Thought,p. 92. In Art 166 I show that undera

certain condition must be single within a certain partof

o

the universe When no condition is imposed upon it, its

meaningis quiteindeterminate.

75 DefinitionoftheIndex sign.

xm> where m is an integral symbol, is defined to mean

theselective operationxrepeated mtimes Forinstance,

x2=xx

Itis to be noted thatx may be single orcomplex, and

positive ornegative or both.

xmxn = x m+n.

I I ocr

VI RULE OF SIGNS.

77 If+2be defined to be equivalent to +,it follows

bytheusual proof, that

H =

j

and

6 Rule of Signs.

Observation It is, of course, immaterial in what con

nection -| occurs; whetherin +( x\or +x( y), or

. The superposition of thetwo mental operations can

be considered apart from the accidents with which theoperationsare mixed up inuse

VII INTEGRAL SYMBOLS.

78 Tofind the meaning of , where m and nareeach

Then nw=m. (Art. 72); for every

definition applies conversely

Since m and n are each integral (that is,the wholere

peated a number of times), it is evident that the latter

equation cannot be satisfied unless n is a divisor of

m Then w = quotient. Thus is impossible unless n

Observation \ does not mean the same as J. For \

means thewhole upon two wholes; whereas J simplymeans

Integral Symbols. 2 7

o

n

Then &w=n (bydefinition).

Suppose w to be integral; the latter equation is then

evidently impossible

Suppose it to be fractional and single; the equation is

then also impossible So also, when w is supposed frac

tionalandcomplex.

Hence isanimpossible selective operation,

o

COR Jfx= y,and xispossible;theny mustbe o

butxbeing apossible selective symbol, 0^=0;

80 Tofindthemeaningof

o

symbol; also when w is any fractional symbol, whether

single orcomplex Hence - isquite indeterminate

81 Since the symbols and the signs of the Algebra of

Quality have a meaning, which is a generalised form of

28 On the Equation

conditions which have beenintroduced are removed For example:

be a a2