Principia Mathematica - Whitehead & Bertrand Russell. Pag 41-60

8/2/2019 Principia Mathematica - Whitehead & Bertrand Russell. Pag 41-60

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I] REAL VARIABLES 19we can only legitimately assert" any value" if all values are true; for other-wise, since the value of tbe variable remains to be determined, it might be sodetermined as to give a false proposition. Thus in the above instance, sincewe havewe may infer

I - .x=xI- (x ) x =x .

And generally, given an assertion containing a real variable ai, we may trans-form the real variable into an apparent one by placing the x in bracketsat the beginning, followed by as many dots as there are after tbe assertion-SIgn.

When we assert something containing a real variable, we cannot strictlybe said to be asserting a proposition, for we only obtain a definite propositionby assigning a value to the variable, and then our assertion only applies toone definite case, so that it has not at all the same force as before. Whenwhat we assert contains a real variable, we are asserting a wholly undeter-mined one of all tbe propositions that result from giving various values tothe variable. Itwill be convenient to speak of such assertions as asserting apropositional function. The ordinary formulae of mathematics contain suchassertions; for example

" sin" to + cos" X =1"does not assert this or tbat particular case of the formula, nor does it assertthat the formula holds for all possible values of x, though it is equivalent tothis latter assertion; it simply asserts that the formula bolds, leaving xwholly undetermined; and it is able to do this legitimately, because, howevera : may be determined, a true proposition results.

Although an assertion containing a real variable does not, in strictness,assert a proposition, yet it will be spoken of as asserting a proposition exceptwhen the nature of the ambiguous assertion involved is under discussion.

Definition and real variables. When the definiells contains one or morereal variables, the definiendum must also contain them. For in this case wehave a function of the real variables, and the definiendum must have the samemeaning as the definiens for all values of these variables, which requires thatthe symbol which is the definiendum should contain the letters representingthe real variables. This rule is not always observed by mathematicians, andits infringement has sometimes caused important confusions of thought,notably in geometry and the philosophy of space.

In the definitions given above of "p q" and "p : : > q" and "p = = q," p and qare real variables, and therefore appear on both sides of the definition. Inthe definition of "", { (x ) c f> x}" only the function considered, namely c f > ~ , is a realvariable; thus so far as concerns the rule in question, x need not appear onthe left. But when a real variable is a function, it is necessary to indicate

2-2


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20 INTRODUCTION [CHAP,how the argument is to be supplied, and therefore there are objections toomitting an apparent variable where (as in the case before us) this is theargument to the function which is the real variable. This appears moreplainly if, instead of a general function cp2V,we take some particular function,say "2 V=a," and consider the definition of "" {(x) x =a}. Our definitiongIves '" !(x). to=a} =. (~x) "" (x = ) Df.But if we had adopted a notation in which the ambiguous value" o : =a ,"containing the apparent variable to, did not occur in the definiendum, weshould have had to construct a notation employing the function itself,namely "2 V= a ." This does not involve an apparent variable, but would beclumsy in practice. In fact we have found it convenient and possible-exceptin the explanatory portions-to keep the explicit use of symbols of the type" c p2V ,"either as constants [e .g . 2 V=a ] or as real variables, almost entirely outof this work.

Propositions connecting rea l and apparen t variab les. The most importantpropositions connecting real and apparent variables are the following:

(1) "When a propositional function can be asserted, so can the propo-sition that all values of the function are true." More briefly, if less exactly," what holds of any, however chosen, holds of all." This translates itself intothe rule that when a real variable occurs in an assertion, we may turn it intoan apparent variable by putting the letter representing it in bracketsimmediately after the assertion-sign.

(2) "What holds of all, holds of any," i.e.I - : (x) . cpx : : > cpy~

This states" if cpx is always true, then cpy is true."(3) " If cpy is true, then cpx is sometimes true," i.e.

I - : cpy J . (~ x) c px.An asserted proposition of the form "(~ x). cpx" expresses an "existence-theorem," namely "there exists an a : for which x is true." The above pro-position gives what is in practice the only way of proving existence-theorems:we always have to find some particular y for which cpy holds, and thence toinfer "(~x). x ." If we were to assume what is called the multiplicativeaxiom, or the equivalent axiom enunciated by Zermelo, that would, in animportant class of cases, give an existence-theorem where no particularinstance of its truth can be found.

In virtue of " I- : (x) x. : : > cpy" and "I-: y. J (~x) cpx," we have" I - : (x). cpx . J . (~ x ) c px ," i.e. "what is always true is sometimes true."This would not be the case if nothing existed; thus our assumptions containthe assumption that there is something. This is involved in the principle


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I] FORMAL IMPLICATION 21that what holds of all, holds of any; for this would not be true if there wereno "any."

(4) "If x is always true, and 'o/x is always true, then ' x . 'o /x ' is alwaystrue," i .e . I - : . (x ) x : (x ) ' o /x : : : > (x ) x 'o /x .(This requires that and '0 / should be functions which take arguments of thesame t ype . We shall explain this requirement at a later stage.) The conversealso holds; i .e . we have

I - : . (x ) x. 'o /x . : : > : (x ) c px : (x ) . 'o /x .It is to some extent optional which of the propositions connecting real

and apparent variables are taken as primitive propositions. The primitivepropositions assumed, on this subject, in the body of the work (*9), are thefollowing:

(1)(2)

I - : cp x . : : > ( 3:z) c pz.I - : cpx v c p y : : > (3 :z) c pz,

i .e . if either cpx is true, or y is true, then (3:z) . cp z is true. (On thenecessity for this primitive proposition, see remarks on *9'11 in the bodyof the work.)

(3) If we can assert y , where y is a real variable, then we can assert(x ) . x ; i.e . what holds of any, however chosen, holds of all.

Fo rm al im plica tio n a n d formal equ i va l ence . When an implication, say x . : : > 'o/x , is said to hold always, i .e . when (x ) : cpx . : : > 'o/x , we shall say thatx fo rm ally im plie s 'o /m ; and propositions of the form" (x ) : c px : : > 'o/x" willbe said to state fo rm a l im p lica tio ns . In the usual instances of implication,such as '" Socrates is a man' implies' Socrates is mortal,''' we have a propo-sition of the form" x . : : > 'o/x" in a case in which" (x ) : x . : : > tx " is true.In such a case, we feel the implication as a particular case of a formal impli-cation. Thus it has come about that implications which are not particularcases of formal implications have not been regarded as implications at all.There is also a practical ground for the neglect of such implications, for, speakinggenerally, they can only be known when it is already known either that theirhypothesis is false or that their conclusion is true; and in neither of thesecases do they serve to make us know the conclusion, since in the first casethe conclusion need not be true, and in the second it is known already.Thus such implications do not serve the purpose for which implications arechiefly useful, namely that of making us know, by deduction, conclusions ofwhich we were previously ignorant. Fo rma l implications, on the contrary,do serve this purpose, owing to the psychological fact that we often know"(x ) : x . : : > 'o/x" and c py , in cases where 'o/Y (which follows from thesepremisses) cannot easily be known directly.


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22 INTRODUCTION [CHAP.These reasons, though they do not warrant the complete neglect of impli-

cations that are not instances of formal implications, are reasons which makeformal implication very important. A formal implication states that, for allpossible values of ai, if the hypothesis c f>x is true, the conclusion Vx is true.Since" c P x : : > Vx" will always be true when c f>xis false, it is only the valuesof a : that make cpx true that are important in a formal implication; what iseffectively stated is that, for all these values, " " " x is true. Thus propositionsof the form "all IX is / 3 , " "no IX is / 3 " state formal implications, since thefirst (as appears by what has just been said) states

(x) : x is an Il: > x is a / 3 ,while the second states

(x) : x is an Il: > a: is not a / 3 .And any formal implication" (x ) : c f> x . : : > vx" may be interpreted as: "Allvalues of x which satisfy * c f>x satisfy vx," while the formal implication"(x ) : c f>x . : : > ." 'Vx" may be interpreted as: "No values of x which satisfy cf>xsatisfy Vx."

We have similarly for" some IX is / 3 " the formula(~x) x is an IX . x is a / 3 ,

and for" some IX is not / 3 " the formula(~x) x is an IX o : is not a / 3 .

Two functions x , Vx are called fo rm ally equivalen t when each alwaysimplies the other, i .e. when

(x) : cf> x. = = Vx,and a proposition of this form is called a fo rm al equ iv ale nce . In virtue ofwhat was said about truth-values, if c f>xand vx are formally equivalent, eithermay replace the other in any truth-function. Hence for all the purposes ofmathematics or of the present work, c f>~may replace V~ or vice versa in anyproposition with which we shall be concerned. Now to say that c f>xand "irxare formally equivalent is the same thing as to say that c f>~and V~have thesame extension, i.e . that any value of x which satisfies either satisfies theother. Thus whenever a constant function occurs in our work, the truth-value of the proposition in which it occurs depends only upon the extensionof the function. A proposition containing a function c f>~and having thisproperty (i.e. that its truth-value depends only upon the extension of c f > ~ ) willbe called an extensional function of c f>~. Thus the functions of functions withwhich we shall be specially concerned will all be extensional functions offunctions.

What has just been said explains the connection (noted above) betweenthe fact that the functions of propositions with which mathematics is specially

* A value of x is said to satisfy x or x when x is true for that value of x.


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I] IDENTITY 23concerned are all truth-functions and the fact that mathematics is concernedwith extensions rather than intensions.

Co n ve n ie n t a b b re via tio n . Tbe following definitions give alternative andoften more convenient notations:

x. ::> x .y .x : = : (x ): x . : : > . y .x Df,x . ==x .y .x : =: (x ) : c f> x. = = y .x Df.

This notation " x . ::> x y .x" is due to Peano, who, however, has no notationfor the general idea" (x ) x ." Itmay be noticed as an exercise in the useof dots as brackets that we might have written

x ::> xtx .=.(x ) . x ::> tx Df, x ==x .x . =.(x ) . x = = y .x Df.

In practice however, when fi: and 'tiC are special functions, it is not possibleto employ fewer dots than in the first form, and often more are required.The following definitions give abbreviated notations for functions of twoor more variables:

(x , y ) (x , y ) =: (x ) : (y ). (x , y ) Df,and so on for any number of variables;

(x , y ) ::> x,y y . (x , y ) : = : (x , y ) : (x , y ) : : > y . (x , y) Df,and so on for any number of variables.

Ident i ty . The propositional function "x IS identical with y" IS ex-pressed by x=y .This will be defined (cf. *13'01), but, owing to certain difficult points involvedin the definition, we shall here omit it (cf. Chapter II). We have, ofcourse, r . X= ill (the law of identity),

r : x =y . = = . y =x,r : o : = y y = z . : : > x =z.

The first of these expresses the re f lex ive property of identity: a relation iscalled re f lee iue when it holds between a term and itself, either universally, orwhenever it holds between that term and some term. The second of theabove propositions expresses that identity is a symme t r i ca l relation: a relationis called symme t r i ca l if, whenever it holds between x and y , it also holdsbetween y and a : The third proposition expresses that identity is a t rans i t iverelation: a relation is called t rans i t ive if, whenever it holds between x and yand between y and z, it holds also between a: and e.

We shall find that no new definition of the sign of equality is required inmathematics: all mathematical equations in which the sign of equality is

/


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24 INTRODUCTION [CHAP.used in the ordinary way express some identity, and thus use the sign ofequality in the above sense.

If x and yare identical, either can replace the other in any propositionwithout altering the truth-value of the proposition; thus we have

I- : a : =y : : > c p x = = c p y .This is a fundamental property of identity, from which the remaining propertiesmostly follow.

It might be thought that identity would not have much importance, sinceit can only hold between a : and y if a: and yare different symbols for thesame object. This view, however, does not apply to what we shall call" descripti ve phrases," i.e. "the so-and-so." It is in regard to such phrasesthat identity is important, as we shall shortly explain. A proposition suchas "Scott was the author of Waverley" expresses an identity in which thereis a descriptive phrase (namely 'e the author of Waverley"); this illustrateshow, in such cases, the assertion of identity may be important. It isessentially the same case when the newspapers say "the identity of thecriminal has not transpired." In such a case, the criminal is known by adescriptive phrase, namely "the man who did the deed," and we wish tofind an x of whom it is true that" x =the man who did the deed." Whensuch an a : has been found, the identity of the criminal has transpired.

Classes and rela tions. A class (which is the same as a mani fo ld oraggregate ) is all the objects satisfying some propositional function. If a isthe class composed of the objects satisfying c p : 0 , we shall say that a is the classde te rmined by cp:0. Every propositional function thus determines a class,though if the propositional function is one which is always false, the classwill be null, i.e. will have no members. The class determined by the functioncp:0will be represented by : Z (cpz)*. Thus for example if cpx is an equation,: Z (cpz) will be the class of its roots; if cpx is" x has two legs and no feathers,": Z (cpz) will be the class of men; if cpx is "0 < to< 1," : Z (cpz) will be the classof proper fractions, and so on.

It is obvious that the same class of objects will have many determiningfunctions. When it is not necessary to specify a determining function of aclass, the class may be conveniently represented by a single Greek letter.Thus Greek letters, other than those to which some constant meaning isassigned, will be exclusively used for classes.

There are two kinds of difficulties which arise in formal logic; one kindarises in connection with classes and relations and the other in connectionwith descriptive functions. The point of the difficulty for classes andrelations, so far as it concerns classes, is that a class cannot be an' objectsuitable as an argument to any of its determining functions. If a represents

* Any other letter may be used instead of z,


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J] CLASSES 25a class and e p i C one of its determining functions [so that a = : t ( e p z ) ] , it is notsufficient that a be a false proposition, it must be nonsense. Thus acertain classification of what appear to be objects into things of essentiallydifferent types seems to be rendered necessary. This whole question is dis-cussed in Chapter II, on the theory of types, and the formal treatment in thesystematic exposition, which forms the main body of this work, is guidedby this discussion. The part of the systematic exposition which is speciallyconcerned with the theory of classes is *20, and in this Introduction it isdiscussed in Chapter III. It is sufficient to note here that, in the completetreatment of *20, we have avoided the decision as to whether a class ofthings has in any sense an existence as one object. A decision of thisquestion in either way is indifferent to our logic, though perhaps, if we hadregarded some solution which held classes and relations to be in some realsense objects as both true and likely to be universally received, we mighthave simplified one or two definitions and a few preliminary propositions.Our symbols, such as "i C ( e p ~ ) " and a and others, which represent classesand relations, are merely defined in their use, just as V2, standing for

( 1 2 ( 1 2 02-+-+---O X 2 o y 2 O Z 2 'has no meaning apart from a suitable function of c, y, z on which to operate.The result of our definitions is that the way in which we use classes corre-sponds in general to their use in ordinary thought and speech; and whatevermay be the ultimate interpretation of the one is also the interpretation ofthe other. Thus in fact our classification of types in Chapter II reallyperforms the single, though essential, service of justifying us in refrainingfrom entering on trains of reasoning which lead to contradictory conclusions.The justification is that what seem to be propositions are really nonsense.

The definitions which occur in the theory of classes, by which the idea ofa class (at least in use) is based on the other ideas assumed as primitive,cannot be understood without a fuller discussion than can be given now(cf. Chapter II of this Introduction and also *20). Accordingly, in thispreliminary survey, we proceed to state the more important simple pro-positions which result from those definitions, leaving the reader to employ inhis mind the ordinary unanalysed idea of a class of things. Our symbolsin their usage conform to the ordinary usage of this idea in language.It is to be noticed that in the systematic exposition our treatment of classesand relations requires no new primitive ideas and only two new primitivepropositions, namely the two forms of the" Axiom of Reducibility" (cf. nextChapter) for one and two variables respectively.

The propositional function "x is a member of the class a" will beexp~essed, following Peano, by the notation

xea.


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26 INTRODUCTION [CHAP.Here e is chosen as the initial of the word ( T T t . x e ex" may be read" a; isan ex." Thus" [JJ e man" will mean" x is a man," and so on. For typographicalconvenience we shall put

X r-.J e e x =.r-.J (x e e x ) Df,ai, y ex.=.X ex. y e ex Df.

For" class" we shall write C l s " ; thus" e x CIs" means" exis a class."We have

I - : x 'Z (z) . = = x,t.e. "'x is a member of the class determined by 'Z' is equivalent to " a :satisfies 'Z,' or to 'x is true.'"

A class is wholly' determinate when its membership is known, that is,there cannot be two different classes having the same membership. Thus ifx, y.x are formally equivalent functions, they determine the same class;for in that case, if x is a member of the class determined by f i f , and thereforesatisfies x, it also satisfies 1[rx, and is therefore a member of the classdetermined by y . f i f . Thus we have

I - : . 'Z (cpz)=Z (y.z) = = : x. ==x' 1[rx.The following propositions are obvious and important:

I - : . e x = Z ( z) = = : [JJ e e x = = x x,s.e, ex is identical with the class determined by 'Z when, and only when,"x is an ex" is formally equivalent to x;

I - : . ex=f J = = : x e x = = x X f J ,~.e. two classes exand f J are identical when, and only when, they have thesame membership;

I - f if (x e x ) =e x ,i .e . the class whose determining function is "[JJ is an e x " is e x , in other words,e x is the class of objects which are members of e x ;

I- 'Z (z) CIs,i .e. the class determined by the function 'Z is a class.Itwill be seen that, according to the above, any function of one variablecan be replaced by an equivalent function of the form "x e ex." Hence any

extensional function of functions which holds when its argument is a functionof the form" 'Z e ex,"whatever possible value exmay have, will hold also whenits argument is any function 'Z. Thus variation of classes can replacevariation of functions of one variable III all the propositions of the sort withwhich we are concerned.

In an exactly analogous manner we introduce dual or dyadic relations,i .e . relations between two terms. Such relations will be called simply" relations"; relations between more than two terms will be distinguished


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I] RELATIONS 27as multiple relations, or (when the number of their terms is specified) astriple, quadruple, ... relations, or as triadic, tetradic, ... relations. Such relationswill not concern us until we come to Geometry. For the present, the onlyrelations we are concerned with are dual relations.

Relations, like classes, are to be taken in extension, i.e. if Rand S arerelations which hold between the same pairs of terms, Rand S are to beidentical. We may regard a relation, in the sense in which it is requiredfor our purposes, as a class of couples; i.e. the couple (x, y) is to be one ofthe class of couples constituting the relation R if o : has the relation R to y*.This view of relations as classes of couples will not, however, be introducedinto our symbolic treatment, and is only mentioned in order to show that itis possible so to understand the meaning of the word relation that a relationshall be determined by its extension.

Any function cp (x, y) determines a relation R between x and y. If weregard a relation as a class of couples, the relation determined by c p (x, y) isthe class of couples (x, y) for which (x, y) is true. The relation determinedby the function (x, y) will be denoted by

5 3y cp (x , y ).We shall use a capital letter for a relation when it is not necessary tospecify the determining function. Thus whenever a capital letter occurs, itis to be understood that it stands for a relation.

The propositional function "to has the relation R to y" will be expressedby the notation

xRy .-This notation is designed to keep as near as possible to common language,which, when it has to express a relation, generally mentions it between itsterms, as in " x loves y," "x equals y," "x is greater than y, " and so on. For" relation" we shall write" ReI"; thus" R ReI" means" R is a relation."

Owing to our taking relations in extension, we shall havef - : . 53y (x, y) =53yt' (x, y) = = : cp (x, y) =x,y t'(z, y) ,

i.e. two functions of two variables determine the same relation when, andonly when, the two functions are formally equivalent.

We have f- z 153ycp(x , y ) } w . = = . cp (z, w),i.e. "z has to w the relation determined by the function cp (x, y)" is equivalentto cp (z, w);

f - : . R =53y (x, y). = = : xRy . ==x,y. cp (x, y) ,f - : . R = S = = : xRy . ==x,y xSy,f - 53y(xRy) =R ,f - {53ycp(z, y) } ReI.

* Such a couple has a s ense , i,e. the couple (x , y) is different from the couple (y , x ), unlessx=y. We shall call it a "couple with sense," to distinguish it from the class consisting of xand y. Itmay also b~ called an ordered couple.


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28 INTRODUCTION [CHAP.These propositions are analogous to those previously given for classes.

Itresults from them that any function of two variables is formally equivalentto some function of the form xRy; hence, in extensional functions of twovariables, variation of relations can replace variation of functions of twovariables.

Both classes and relations have properties analogous to most of those ofpropositions that result from negation and the logical sum. The logicalproduct of two classes a and (3 is their common part, i.e. the class of termswhich are members of both. This is represented by an (3 . Thus we put

an (3 =$ J ( x a x e (3 ) Df.This gives us I- : x e x n / 3 . = = . X e e x x e (3 ,

i.e. "x is a member of the logical product of a and (3 " is equivalent to thelogical product of" a: is a member of ex"and" ai is a member of (3."

Similarly the logical sum of two classes a and (3 is the class of termswhich are members of either; we denote it by a v (3 . The definition isa v / 3 =$ J (x ex. v . X (3 ) Df,

and the connection with the logical sum of propositions is given byI- :. x exv (3 = = : m a v X (3 .

The negation of a class a consists of those terms x for which" x s a" canbe significantly and truly denied. We shall find that there are terms of othertypes for which" a: s a" is neither true nor false, but nonsense. These termsare not members of the negation of ex.

Thus the negation of a class a is the class of terms of suitable typewhich are not members of it, i.e. the class $ J ( x " - ' e x ) . We call this class "-a"(read" not-a "); thus the definition is

- a = $ J ( x " -' ex) Df,and the connection with the negation of propositions is given by

I- : x - ex. = = . X"-' a.In place of implication we have the relation of inclusion. A class ex

IS said to be included or contained in a class (3 if all members of a aremembers of (3 , i.e. if o : a . : : : > a ; X e (3 . We write" exC f1 " for" a is containedin / 3 . " Thus we put

a C / 3 . =:x a : : : > a ; X e (3 Df.Most of the formulae concerning P > q , p v q , "-'p , p : : : > q remain true if wesubstitute a n / 3 , a v (3, - a, a C (3 . In place of equivalence, we substitute

identity; for IIp = = q" was defined as "p : : : > q q : : : > p ," but" a C (3. (3 C a" gives"ee a : = = o t: . X / 3 , " whence a : = 3 .


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CALCULUS OF CLASSES 29The following are some propositions concerning classes which are analogues

of propositions previously given concerning propositions:f - a " ,8=- (- a v - ,8),

i.e. the common part of a and ,8 is the negation of" not-a or not-,8" ;f - X (a v - a ),i.e. "x is a member of a or not-a";

f - x", (a" - a),i.e. "x is not a member of both a and not-a" ;

I - a =- (- a ) ,I - : a C,8 . = = . - , 8C- a,f - : a = / 3 = = . - a = - ,8,I - : a =a " a,f - : a = a v a .

The two last are the two forms of the law of tautology.The law of absorption holds in the form

f - : ' : ; (C,8 . = = . a =a n ,8.Thus for example" all Cretans are liars" is equivalent to " Cretans are

identical with lying Cretans."Just as we have f - : p J q . q Jr. J .P J r,

so we have f - : 11 C,8 . ,8 C "/ . J . a C"/.This expresses the ordinary syllogism in Barbara (with the premissesinterchanged); for" a C,8" means the same as "all a's are ,8's," so that the

above proposition states: "If all a's are ,8's, and all ,8's are ,,/'s, then all a'sare "/'s." (It should be observed that syllogisms are traditionally expressedwith" therefore," as if they asserted both premisses and conclusion. This is,of course, merely a slipshod way of speaking, since what is really asserted isonly the connection of premisses with conclusion.)

The syllogism III Barbara when the minor premiss has an individualsubject is

I - : x ,8 . ,8 C"/ . J . X "/,e .g. "if Socrates is a man, and all men are mortals, then Socrates is amortal." This, as was pointed out by Peano, is not a particular case of" 11 C,8 ,8 C "/ . J . a C ,,/," since "x ,8" is not a particular case of "a C,8."This point is important, since traditional logic is here mistaken. The natureand magnitude of its mistake will become clearer at a later stage.

For relations, we have precisely analogous definitions and propositions.We putwhich leads to

R;.. S=a ; f ) (xRy . xSy) Df,f - : x (R n S) y = = . xRy xSy.


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30 INTRODUCTION [CHAP.R \;J S = f e y (xRy V xSy) Df,.:...R= e y {r-.;(xRy)} Df,

R c : : S =:xRy . J ; x , y xSy Df.Generally, when we require analogous but different symbols for relations

and for classes, we shall choose for relations the symbol obtained by addinga dot, in some convenient position, to the corresponding symbol for classes.(The dot must not be put on the line, since that would cause confusion withthe use of dots as brackets.) But such symbols require and receive a specialdefinition in each case.

Similarly

A class is said to exist when it has at least one member: "a exists"is denoted by "~! " Thus we put

~! a.. =. (~x). tc e a Df.The class which has no members is called the "null-class," and isdenoted by "A." Any propositional function which is always false deter-mines the null-class. One such function is known to us already, namely"x is not identical with x," which we denote by "x = + = ai" Thus we may usethis function for defining A, and put

A =f e (x = + = x) Df.The class determined by a function which is always true is called theuniversal class, and is represented by V; thus

V =~(x = x) Df.Thus A is the negation of V. We have

1 - . (x). X V,i.e. '" x is a member of V' is always true"; and

I- tx) . XC'.) A,i.e. "'x is a member of A' is always false." Also

I - : II=A . = = . r-.J ~ !a,i.e. "a. is the null-class" is equivalent to "a. does not exist."

For relations we use similar notations. We put3:! R. =.(~x, y). xRy,

i.e. "3:! R" means that there is at least one couple ta, y between which therelation R holds. A will be the relation which never holds, and V therelation which always holds. V is practically never required; A will be therelation f e y (x = + = a: y = f y) . We have

I - (x,y).r-.;(xAy),1 - : R=A . = = . "'3:!R.nd


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I] DESCRIPTIONS 31There are no classes which contain objects of more than one type.

Accordingly there is a universal class and a null-class proper to each typeof object. But these symhols need not be distinguished, since it will befound that there is no possibility of confusion. Similar remarks apply torelations.

Descr ip t ions . By a "description" we mean a phrase of the form" th eso-and-so" or of some equivalent form. For the present, we confine ourattention to th e in the singular. We shall use this word strictly, so as toimply uniqueness; e .g . we should not say" A is th e son of B" if B had othersons besides A. Thus a description of the form" the so-and-so" will onlyhave an application in the event of there being one so-and-so and no more.Hence a description requires some propositional function c p a ; which is satisfiedby one value of a : and by no other values; then "the a: which satisfies c p a ; "is a description which definitely describes a certain object, though we maynot know what object it describes. For example, if y is a man, "x is thefather of y" must be true for one, and only one, value of a: Hence" thefather of y" is a description of a certain man, though we may not know wha tman it describes. A phrase containing "the" always presupposes someinitial propositional function not containing" the"; thus instead of" o : is thefather of y" we ought to take as our initial function" to begot y"; then" thefather of y" means the one value of a : which satisfies this propositionalfunction.If c p a ; is a propositional function, the symbol " ( 1 X ) ( c p X ) " is used in our

symbolism in such a way that it can always be read as "the a : whichsatisfies c p a ; . " But wo do not define " ( 1 a ; ) ( c p x ) " as standing for "the towhich satisfies c p a ; , " thus treating this last phrase as embodying a primitiveidea. Every use of " ( 1 X ) ( c p X ) , " where it apparently occurs as a constituentof a proposition in the place of an object, is defined in terms of the primitiveideas already on hand. An example of this definition in use is given bythe proposition "E! ( 1 X ) ( c p X ) " which is considered immediately. The wholesubject is treated more fully in Chapter III.

The symbol should be compared and contrasted with " a ; ( c p x ) " which inuse can always be read as "the x 's which satisfy c p a ; . " Both symbols areincomplete symbols defined only in use, and as such are discussed inChapter III. The symbol " a ; ( c p x ) " always has an application, namely tothe class determined by c p x ; but" ( 1 X ) ( c p X ) " only has an application whenc p a ; is only satisfied by one value of x , neither more nor less. It should alsobe observed that the meaning given to the symbol by the definition, givenimmediately below, of E! ( 1 X ) ( c p X ) does not presuppose that we know themeaning of "one." This is also characteristic of the definition of any otheruse of ( l X ) ( c p X ) .


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3 2 INTRODUCTION [CHAP.We now proceed to define "E!(1J:)(epx)" so that it can be read "the x

satisfying epx exists." (It will he observed that this is a different meaningof existence from that which we express by :l"[.") Its definition is

E!(lX)(epX).=:(~c):epx.==x'x=c Df,i.e. "the x satisfying epee exists" is to mean "there is an object c suchthat epx is true when x is c but not otherwise."

The following are equivalent forms:~:. E! (lX) (epx) = = : (~c): epc: epx. Jx x = c,~:. E! (lX) (e px ). == : (~c). epc: epx. epy Jx,y' to =y,~:.E! (lX)(epX). = = : (~c): epc: x4=c. Jx . . . ._,epx.

The last of these states that" the x satisfying epeeexists" is equivalent to" there is an object c satisfying epee,and every object other than c does notsatisfy epfiJ."The kind of existence just defined covers a great many cases. Thusfor example" the most perfect Being exists" will mean:

(~c) : x is most perfect. ==x X =c,which, taking the last of the above equivalences, is equivalent to

(~c ) : c is most perfect: to 4 = c Jx x is not most perfect.A proposition such as "Apollo exists" is really of the same logical form,

although it does not explicitly contain the word the. For" Apollo" meansreally" the object having such-and-such properties," say" the object havingthe properties enumerated in the Classical Dictionary*." Ifthese propertiesmake up the propositional function epx, then "Apollo" means "( l X) ( ep x) ,"and "Apollo exists" means "E! ( lX ) ( ep x) ." To take another illustration,"the author of Waverley" means "the man who (or rather, the objectwhich) wrote Waverley." Thus" Scott is the author of Waverley" is

Scott = (lX) (x wrote Waverley).Here (as we observed before) the importance of identity in connection withdescriptions plainly appears.

The notation "(lX) (e px )," which is long and inconvenient, is seldom used,being chiefly required to lead up to another notation, namely" R'y," meaning"the object having the relation R to y." That is, we put

R 'y =(ue) (xRy) Df.The inverted comma may be read" of." Thus" R'y" is read" the R of y."Thus if R is the relation of father to son, " R'y" means" the father of y";if R is the relation of son to father, " R'y" means It the son of y," which will

* The same principle applies to many uses of the proper names of existent objects, e.g. to alluses of proper names for objects known to the speaker only by report, and not by personalaoquaintance.


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I] DESCRIPTIVE FUNCTIONS 3 3only" exist" if y has one son and no more. R'y is a function of y, but nota propositional function; we shall call it a descriptive function. All theordinary functions of mathematics are of this kind, as will appear more fullyin the sequel. Thus in our notation, "sin y" would be written" sin 'y," and"sin" would stand for the relation which sin 'y has to y. Instead of a variabledescriptive function fy, we put R'y, where the variable relation R takesthe place of the variable function f A descriptive function will in generalexist while y belongs to a certain domain, but not outside that domain;thus if we are dealing with positive rationals, ";y will be significant if yis a perfect square, but not otherwise; if we are dealing with real numbers,and agree that "";y" is to mean the positive square root (or, is to mean thenegative square root), ";y will be significant provided y is positive, but nototherwise; and so on. Thus every descriptive function has what we maycall a "domain of definition" or a "domain of existence," which may be thusdefined: If the function in question is R'y, its domain of definition or ofexistence will be the class of those arguments y for which we have E! R'y,i.e. for which E! ( 1 X ) (xRy), i.e. for which there is one ai, and no more, havingthe relation R to y.

If R is any relation, we will speak of R'y as the "associated descriptivefunction." A great many of the constant relations which we shall haveoccasion to introdnce are only or chiefly important on account of theirassociated descriptive functions. In such cases, it is easier (though lesscorrect) to begin by assigning the meaning of the descriptive function, andto deduce the meaning of the relation from that of the descriptive function.This will be done in the following explanations of notation.

Various descriptive functions of relations. If R is any relation, theconverse of R is the relation which holds between y and x whenever Rholds between x and y. Thus greater is the converse of less, before ofafter, cause of effect, husband of wife, etc. The converse of R is written *

vOnv'R or R. The definition isv

vOnv'R=R Df.

The second of these is not a formally correct definition, since we ought todefine" Onv" and deduce the meaning of Onv'R. But it is not worthwhile to adopt this plan in our present introductory account, which aimsat simplicity rather than formal correctness.

vA relation is called symmetrical if R = R, i.e. if it holds between y and a :whenever it holds between x and y (and therefore vice versa). Identity,

* The second of these notations is taken from Schroder's Alqebra uud Logik der Relative.R. & w. 3


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34 INTRODUCTION [CHAP.diversity, agreement or disagreement in any respect, are symmetrical relations.A relation is called asymmetrical when it is incompatible with its converse,

v i .e. when R n R=A, or, what is equivalent,xRy. : : l x , y ,,-,(yRx).

Before and after, greater and less, ancestor and descendant, are asym-metrical, as are all other relations of the sort that lead to series. But there aremany asymmetrical relations which do not lead to series, for instance, that ofwife's brother*. A relation may be neither symmetrical nor asymmetrical;for example, this holds of the relation of inclusion between classes: a C / 3 and/ 3 C a will both be true if a = 3 , but otherwise only one of them, at most, willbe true. The relation brother is neither symmetrical nor asymmetrical, forif x is the brother of y, y may be either the brother or the sister of a :

In the propositional function xRy, we call a : the referent and y therelatum. The class ~ (xRy), consisting of all the x's which have the relationR to y, is called the class of referents of y wl'th respect to xrb_the class ~p (xRy), consisting of all the y's to which x has the relation R, is called theclass of relata of x with respect to R. These two classes are denoted~ ~respectively by R'y and R'x. Thus

~R'y = ~ (xRy) Df,~ ':/ VR'x= P(yRx) Df.The arrow runs towards y in the first case, to show that we are concerned

with things having the relation R to y; it runs away from x in the second~case to show that the relation R goes from o : to the members of R'x.It runs in fact from a referent and towards a relatum.

~ ~The notations R'y, R'x are very important, and are used constantly. If~ ~R is the relation of parent to child, R'y = the parents of y, R'x = the childrenof x, We have

and

_ .I - : a : R'y = = . xRy

~I - : y e R'x . = = . xRy.These equivalences are often embodied in common language. For example,we say indiscriminately" a : is an inhabitant of London" or "x inhabits London."Ifwe put" R" for" inhabits," " inhabits London" is "xR London," while" a:~is an inhabitant of London" is " X R' London."

* This relation is not strictly asymmetrical, but is so except when the wife's brother is alsothe sister's husband. In the Greek Church the relation is strictly asymmetrical.


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I] DOMAINS AND FIELDS 35- -+ +-Instead of Rand R we sometimes use sg'R, gs'R, where" sg" stands for, . "d " . "b k d ThsagItta, an "gs IS" sg ac war s. us we put

-- +sg'R=R Df,+-gs'R=R Df.

These notations are sometimes more convenient than an arrow when therelation concerned is represented by a combination of letters, instead of asingle letter such as R. Thus e .g . we should write sg'(R n S), rather thanput an arrow over the whole length of (R nS).

The class of all terms that have the relation R to something or other iscalled the domain of R. Thus if R is the relation of parent and child, thedomain of R will be the class of parents. We represent the domain of R by"D'R." Thus we put D'R=~{(3:y).xRy} Df.Similarly the class of all terms to which something or other has the relationR is called the converse domain of R; it is the same as the domain of theconverse of R. 'I'he converse domain of R is represented by "G'R"; thus

G'R =Y {(3:x). xRy} Df.The sum of the domain and the converse domain is called the field, and isrepresented by G'R: thus

G'R = D'R v G'R Df.The field is chiefly important in connection with series. If R is theordering relation of a series, G'R will be the class of terms of the series, D'Rwill be all the terms except the last (if any), and G'R will be all the termsexcept the first (if any). The first term, if it exists, is the only member ofD'R" - G'R, since it is the only term which is a predecessor but not afollower. Similarly the last term (if any) is the only member of G'R,,-D'R.The condition that a series should have no end is G'R C D'R, i.e. "everyfollower is a predecessor"; the condition for no beginning is D'R C G'R.These conditions are equivalent respectively to D'R= G'R and G'R =G'R.

The relative product of two relations Rand S is the relation which holdsbetween o: and z when there is an intermediate term y such that a : has therelation R to y and y has the relation S to z. The relative product of RandS is represented by R IS ; thus we put

R I S =~:Z {(3Y). xRy. ySz} Df,whence 1 - : x(R I S)z. = = . (3:Y). xRy. ySz.Thus" paternal aunt" is the relative product of sister and father; "paternalgrandmother" is the relative product of mother and father; "maternal

3-2


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36 INTRODUCTION [CHAP.grandfather" is the relative product of father and mother. The relative pro-duct is not commutative, but it obeys the associative law, i.e.

~ . (P I Q) I R =PI (Q i R).It also obeys the distributive law with regard to the logical addition ofrelations, i.e. we have

~ . P I (Q \:I R) = (P I Q) \:I (P I R),~ . (Q \:IR) I P =(Q I P) \:I (ttt R).

But with regard to the logical product, we have only~ . P I (Q nR) C ! (P I Q) n (P IR),~ . (Q n R) I P C ! (Q I P) n (QtR). "

The relative product does not obey the law of tautology, i.e. we do nothave in general R I R=R. We putR2=RIR Df.Thus paternal grandfather =(father)",

maternal grandmother =(mother)",A relation is called transitive when R2 C ! R, i.e. when, if xRy and yRz, we

always have xRz, i.e. whenxRy. yRz. )x,y,z' xRz.

Relations which generate series are always transitive; thus e .g .x> y Y >z , )x,y,Z x >s.If P is a relation which generates a series, P may conveniently be read

" precedes"; thus" xPy . yPz. )x,y,z' Pe" becomes" if x precedes y and yprecedes z, then to always precedes z." The class of relations which generateseries are partially characterized by the fact that they are transitive andasymmetrical, and never relate a term to itself.

If P is a relation which generates a series, and if we have not merelyP2C!P,but P=P, then P generates a series which is compact (uberall dicht),i.e. such that there are terms between any two. For in this case we have

xPz . ) . (ay) xPy yPz,i.e. if x precedes e, there is a term y such that a : precedes y and y precedes e,i.e. there is a term between x and z, Thus among relations which generateseries, those which generate compact series are those for which P = P.

Many relations which do not generate series are transitive, for example,identity, or the relation of inclusion between classes. Such cases arisewhen the relations are not asymmetrical. Relations which are transitiveand symmetrical are an important class: they may be regarded as consistingin the possession of some common property',


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I] PLURAL DESCRIPTIVE FUNCTIONS 3 7.Plural descriptive functions. The class of terms x which have the relationR to some member of a class a is denoted by R"a. or R.'a. The definition is

R"a. =S : { (:E [ Y ) Y a . xRy} Df.Thus for example let R be the relation of inhabiting, and Il the class oftowns; then R"a. =inhabitants of towns. Let R be the relation" less than"among rationals, and a . the class of those rationals which are of the form1 - 2-n, for integral values of n; then R"1l will be all rationals less thansome member of Il, i.e. all rationals less than l. If P is the generatingrelation of a series, and a is any class of members of P, P"1l will be pre-decessors of a's, i.e. the segment defined by a.. If P is a relation such thatP'y always exists when Y Il, P'! will be the class of all terms of the formP'y for values of y which are members of a.; i.e.

P'! = S : [ ( :E [y ) Y a.. o : = P'y}.Thus a member of the class" fathers of great men" will be the father of y,where y is some great man. In other cases, this will not hold; for instance,let P be the relation of a number to any number of which it is a factor; thenpee (even numbers) = factors of even numbers, but this class is not composedof terms of the form "the factor of x," where x is an even number, becausenumbers do not have only one factor apiece.

Unit classes. The class whose only member is a : might be thought to beidentical with tc, but Peano and Frege have shown that this is not the case.(The reasons why this is not the case will be explained in a preliminary wayill Chapter II of the Introduction.) We denote by "L'to " the class whoseonly member is x: thus l'X=y(y=X) Df,i .e . "t'o:" means" the class of objects which are identical with a:"

The class consisting of to and y will be t'o: v l 'y; the class got by addingx to a class a . will be a . v t 'x; the class got by taking away ai from a class awill be a-t'x. (We write a-{3 as an abbreviation for an -{3.)

It will be observed that unit classes have been defined without referenceto the number 1; in fact, we use unit classes to define the number 1. Thisnumber is defined as the class of unit classes, i.e.

1= a { ( :E [ X ) a . = leX} Dr.I - : . /l 1. = = : ( : E [x ) : y Il =11 Y =a :

From this it appears further that1 - : a 1. = = . E! ( 1 X ) ( x /l),

whence 1 - : ~ ( e p z ) 1. = = . E! ( 1 X ) ( e p x ) ,

This leads to

i.e. " ~ ( e p z ) is a unit class" is equivalent to "the x satisfying e p S : exists."


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38 INTRODUCTION [CHAP. I'OJIfa e 1, t' is the only member of a , for the only member of a is the only

vterm to which a has the relation t. Thus" t 'a" takes the place of "(lX) (cpx) ,"v

if ex stands for : Z (cpz). In practice, " t 'a" is a more convenient notation than"(lX) (cpx)," and is generally used instead of" (lX) (cpx) ."The above account. has explained most of the logical notation employed

in the present work. In the applications to various parts of mathematics,other definitions are introduced; but the objects defined by these laterdefinitions belong, for the most part, rather to mathematics than to logic.The reader who has mastered the symbols explained above will find that anylater formulae can be deciphered by the help of comparatively few additionaldefinitions.

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Principia Mathematica - Whitehead & Bertrand Russell. Pag 41-60

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