Principia mathematica, vol. 1 (of 3)

(3) "If is true, then is sometimes true," i.e. An asserted proposition of the form "" expresses an "existence-theorem," namely "there exists an for which is true." The above proposition gives what is in practice the only way of proving existence-theorems: we always have to find some particular for which holds, and thence to infer "." If we were to assume what is called the multiplicative axiom, or the equivalent axiom enunciated by Zermelo, that would, in an important class of cases, give an existence-theorem where no particular instance of its truth can be found.

In virtue of "" and "," we have "," i.e. "what is always true is sometimes true." This would not be the case if nothing existed; thus our assumptions contain the assumption that there is something. This is involved in the principle that what holds of all, holds of any; for this would not be true if there were no "any."

(4) "If is always true, and is always true, then '' is always true," i.e. (This requires that and should be functions which take arguments of the same type. We shall explain this requirement at a later stage.) The converse also holds; i.e. we have

It is to some extent optional which of the propositions connecting real and apparent variables are taken as primitive propositions. The primitive propositions assumed, on this subject, in the body of the work (*9), are the following: i.e. if either is true, or is true, then ( is true. (On the necessity for this primitive proposition, see remarks on *9·11 in the body of the work.)

Formal implication and formal equivalence. When an implication, say , is said to hold always, i.e. when , we shall say that formally implies ; and propositions of the form "" will be said to state formal implications. In the usual instances of implication, such as "'Socrates is a man' implies 'Socrates is mortal,'" we have a proposition of the form "" in a case in which "" is true. In such a case, we feel the implication as a particular case of a formal implication. Thus it has come about that implications which are not particular cases of formal implications have not been regarded as implications at all. There is also a practical ground for the neglect of such implications, for, speaking generally, they can only be known when it is already known either that their hypothesis is false or that their conclusion is true; and in neither of these cases do they serve to make us know the conclusion, since in the first case the conclusion need not be true, and in the second it is known already. Thus such implications do not serve the purpose for which implications are chiefly useful, namely that of making us know, by deduction, conclusions of which we were previously ignorant. Formal implications, on the contrary, do serve this purpose, owing to the psychological fact that we often know "" and , in cases where (which follows from these premisses) cannot easily be known directly.

These reasons, though they do not warrant the complete neglect of implications that are not instances of formal implications, are reasons which make formal implication very important. A formal implication states that, for all possible values of , if the hypothesis is true, the conclusion is true. Since "" will always be true when is false, it is only the values of that make true that are important in a formal implication; what is effectively stated is that, for all these values, is true. Thus propositions of the form "all is ," "no is " state formal implications, since the first (as appears by what has just been said) states while the second states And any formal implication "(" may be interpreted as: "All values of which satisfy[6] satisfy ," while the formal implication "(" may be interpreted as: "No values of which satisfy satisfy ."

Two functions , are called formally equivalent when each always implies the other, i.e. when and a proposition of this form is called a formal equivalence. In virtue of what was said about truth-values, if and are formally equivalent, either may replace the other in any truth-function. Hence for all the purposes of mathematics or of the present work, may replace or vice versa in any proposition with which we shall be concerned. Now to say that and are formally equivalent is the same thing as to say that and have the same extension, i.e. that any value of which satisfies either satisfies the other. Thus whenever a constant function occurs in our work, the truth-value of the proposition in which it occurs depends only upon the extension of the function. A proposition containing a function and having this property (i.e. that its truth-value depends only upon the extension of will be called an extensional function of . Thus the functions of functions with which we shall be specially concerned will all be extensional functions of functions.

What has just been said explains the connection (noted above) between the fact that the functions of propositions with which mathematics is specially concerned are all truth-functions and the fact that mathematics is concerned with extensions rather than intensions.

Convenient abbreviation. The following definitions give alternative and often more convenient notations: This notation "" is due to Peano, who, however, has no notation for the general idea "." It may be noticed as an exercise in the use of dots as brackets that we might have written In practice however, when and are special functions, it is not possible to employ fewer dots than in the first form, and often more are required.

Identity. The propositional function " is identical with " is expressed by This will be defined (cf. *13·01), but, owing to certain difficult points involved in the definition, we shall here omit it (cf. Chapter II). We have, of course, The first of these expresses the reflexive property of identity: a relation is called reflexive when it holds between a term and itself, either universally, or whenever it holds between that term and some term. The second of the above propositions expresses that identity is a symmetrical relation: a relation is called symmetrical if, whenever it holds between and , it also holds between and . The third proposition expresses that identity is a transitive relation: a relation is called transitive if, whenever it holds between and and between and , it holds also between and .

We shall find that no new definition of the sign of equality is required in mathematics: all mathematical equations in which the sign of equality is used in the ordinary way express some identity, and thus use the sign of equality in the above sense.

It might be thought that identity would not have much importance, since it can only hold between and if and are different symbols for the same object. This view, however, does not apply to what we shall call "descriptive phrases," i.e. "the so-and-so." It is in regard to such phrases that identity is important, as we shall shortly explain. A proposition such as "Scott was the author of Waverley" expresses an identity in which there is a descriptive phrase (namely "the author of Waverley"); this illustrates how, in such cases, the assertion of identity may be important. It is essentially the same case when the newspapers say "the identity of the criminal has not transpired." In such a case, the criminal is known by a descriptive phrase, namely "the man who did the deed," and we wish to find an of whom it is true that " = the man who did the deed." When such an has been found, the identity of the criminal has transpired.

Classes and relations. A class (which is the same as a manifold or aggregate) is all the objects satisfying some propositional function. If is the class composed of the objects satisfying , we shall say that is the class determined by . Every propositional function thus determines a class, though if the propositional function is one which is always false, the class will be null, i.e. will have no members. The class determined by the function will be represented by [7]. Thus for example if is an equation, will be the class of its roots; if is " has two legs and no feathers," will be the class of men; if is "," will be the class of proper fractions, and so on.

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

There are two kinds of difficulties which arise in formal logic; one kind arises in connection with classes and relations and the other in connection with descriptive functions. The point of the difficulty for classes and relations, so far as it concerns classes, is that a class cannot be an object suitable as an argument to any of its determining functions. If represents a class and one of its determining functions [so that ], it is not sufficient that be a false proposition, it must be nonsense. Thus a certain classification of what appear to be objects into things of essentially different types seems to be rendered necessary. This whole question is discussed in Chapter II, on the theory of types, and the formal treatment in the systematic exposition, which forms the main body of this work, is guided by this discussion. The part of the systematic exposition which is specially concerned with the theory of classes is *20, and in this Introduction it is discussed in Chapter III. It is sufficient to note here that, in the complete treatment of *20, we have avoided the decision as to whether a class of things has in any sense an existence as one object. A decision of this question in either way is indifferent to our logic, though perhaps, if we had regarded some solution which held classes and relations to be in some real sense objects as both true and likely to be universally received, we might have simplified one or two definitions and a few preliminary propositions.

Our symbols, such as "" and and others, which represent classes and relations, are merely defined in their use, just as , standing for has no meaning apart from a suitable function of , , on which to operate. The result of our definitions is that the way in which we use classes corresponds in general to their use in ordinary thought and speech; and whatever may be the ultimate interpretation of the one is also the interpretation of the other. Thus in fact our classification of types in Chapter II really performs the single, though essential, service of justifying us in refraining from 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 of a 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 this preliminary survey, we proceed to state the more important simple propositions which result from those definitions, leaving the reader to employ in his mind the ordinary unanalysed idea of a class of things. Our symbols in 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 classes and relations requires no new primitive ideas and only two new primitive propositions, namely the two forms of the "Axiom of Reducibility" (cf. next Chapter) for one and two variables respectively.

The propositional function " is a member of the class " will be expressed, following Peano, by the notation Here is chosen as the initial of the word ἐστί. "" may be read " is an ." Thus "" will mean " is a man," and so on. For typographical convenience we shall put

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

The following propositions are obvious and important: i.e. is identical with the class determined by when, and only when, " is an " is formally equivalent to ; i.e. two classes and are identical when, and only when, they have the same membership; i.e. the class whose determining function is " is an " is , in other words, is the class of objects which are members of ; i.e. the class determined by the function is a class.

It will be seen that, according to the above, any function of one variable can be replaced by an equivalent function of the form "." Hence any extensional function of functions which holds when its argument is a function of the form "," whatever possible value may have, will hold also when its argument is any function . Thus variation of classes can replace variation of functions of one variable in all the propositions of the sort with which 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 as multiple relations, or (when the number of their terms is specified) as triple, quadruple, ... relations, or as triadic, tetradic, ... relations. Such relations will not concern us until we come to Geometry. For the present, the only relations we are concerned with are dual relations.

Relations, like classes, are to be taken in extension, i.e. if and are relations which hold between the same pairs of terms, and are to be identical. We may regard a relation, in the sense in which it is required for our purposes, as a class of couples; i.e. the couple ( is to be one of the class of couples constituting the relation if has the relation to [8]. This view of relations as classes of couples will not, however, be introduced into our symbolic treatment, and is only mentioned in order to show that it is possible so to understand the meaning of the word relation that a relation shall be determined by its extension.

Any function determines a relation between and . If we regard a relation as a class of couples, the relation determined by is the class of couples for which is true. The relation determined by the function will be denoted by We shall use a capital letter for a relation when it is not necessary to specify the determining function. Thus whenever a capital letter occurs, it is to be understood that it stands for a relation.

The propositional function " has the relation to " will be expressed by the notation 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 its terms, as in " loves ," " equals ," " is greater than ," and so on. For "relation" we shall write "Rel" thus "" means " is a relation."

Owing to our taking relations in extension, we shall have i.e. two functions of two variables determine the same relation when, and only when, the two functions are formally equivalent. i.e. " has to the relation determined by the function " is equivalent to ;

These propositions are analogous to those previously given for classes. It results from them that any function of two variables is formally equivalent to some function of the form ; hence, in extensional functions of two variables, variation of relations can replace variation of functions of two variables.

Both classes and relations have properties analogous to most of those of propositions that result from negation and the logical sum. The logical product of two classes and is their common part, i.e. the class of terms which are members of both. This is represented by . Thus we put This gives us i.e. " is a member of the logical product of and " is equivalent to the logical product of " is a member of " and " is a member of ."

The negation of a class consists of those terms for which "" can be significantly and truly denied. We shall find that there are terms of other types for which "" is neither true nor false, but nonsense. These terms are not members of the negation of .