Principia mathematica, vol. 1 (of 3)

(1) Descriptions. By an "incomplete" symbol we mean a symbol which is not supposed to have any meaning in isolation, but is only defined in certain contexts. In ordinary mathematics, for example, and are incomplete symbols: something has to be supplied before we have anything significant. Such symbols have what may be called a "definition in use." Thus if we put we define the use of , but by itself remains without meaning. This distinguishes such symbols from what (in a generalized sense) we may call proper names: "Socrates," for example, stands for a certain man, and therefore has a meaning by itself, without the need of any context. If we supply a context, as in "Socrates is mortal," these words express a fact of which Socrates himself is a constituent: there is a certain object, namely Socrates, which does have the property of mortality, and this object is a constituent of the complex fact which we assert when we say "Socrates is mortal." But in other cases, this simple analysis fails us. Suppose we say: "The round square does not exist." It seems plain that this is a true proposition, yet we cannot regard it as denying the existence of a certain object called "the round square." For if there were such an object, it would exist: we cannot first assume that there is a certain object, and then proceed to deny that there is such an object.

Whenever the grammatical subject of a proposition can be supposed not to exist without rendering the proposition meaningless, it is plain that the grammatical subject is not a proper name, i.e. not a name directly representing some object. Thus in all such cases, the proposition must be capable of being so analysed that what was the grammatical subject shall have disappeared. Thus when we say "the round square does not exist," we may, as a first attempt at such analysis, substitute "it is false that there is an object which is both round and square." Generally, when "the so-and-so" is said not to exist, we have a proposition of the form[29]

By an extension of the above argument, it can easily be shown that is always an incomplete symbol. Take, for example, the following proposition: "Scott is the author of Waverley." [Here "the author of Waverley" is .] This proposition expresses an identity; thus if "the author of Waverley" could be taken as a proper name, and supposed to stand for some object , the proposition would be "Scott is ." But if is any one except Scott, this proposition is false; while if is Scott, the proposition is "Scott is Scott," which is trivial, and plainly different from "Scott is the author of Waverley." Generalizing, we see that the proposition is one which may be true or may be false, but is never merely trivial, like ; whereas, if were a proper name, would necessarily be either false or the same as the trivial proposition . We may express this by saying that is not a value of the propositional function , from which it follows that is not a value of . But since may be anything, it follows that ( is nothing. Hence, since in use it has meaning, it must be an incomplete symbol.

It might be suggested that "Scott is the author of Waverley" asserts that "Scott" and "the author of Waverley" are two names for the same object. But a little reflection will show that this would be a mistake. For if that were the meaning of "Scott is the author of Waverley," what would be required for its truth would be that Scott should have been called the author of Waverley: if he had been so called, the proposition would be true, even if some one else had written Waverley; while if no one called him so, the proposition would be false, even if he had written Waverley. But in fact he was the author of Waverley at a time when no one called him so, and he would not have been the author if every one had called him so but some one else had written Waverley. Thus the proposition "Scott is the author of Waverley" is not a proposition about names, like "Napoleon is Bonaparte"; and this illustrates the sense in which "the author of Waverley" differs from a true proper name.

Thus all phrases (other than propositions) containing the word the (in the singular) are incomplete symbols: they have a meaning in use, but not in isolation. For "the author of Waverley" cannot mean the same as "Scott," or "Scott is the author of Waverley" would mean the same as "Scott is Scott," which it plainly does not; nor can "the author of Waverley" mean anything other than "Scott," or "Scott is the author of Waverley" would be false. Hence "the author of Waverley" means nothing.

It follows from the above that we must not attempt to define "(," but must define the uses of this symbol, i.e. the propositions in whose symbolic expression it occurs. Now in seeking to define the uses of this symbol, it is important to observe the import of propositions in which it occurs. Take as an illustration: "The author of Waverley was a poet." This implies (1) that Waverley was written, (2) that it was written by one man, and not in collaboration, (3) that the one man who wrote it was a poet. If any one of these fails, the proposition is false. Thus "the author of 'Slawkenburgius on Noses' was a poet" is false, because no such book was ever written; "the author of 'The Maid's Tragedy' was a poet" is false, because this play was written by Beaumont and Fletcher jointly. These two possibilities of falsehood do not arise if we say "Scott was a poet." Thus our interpretation of the uses of must be such as to allow for them. Now taking to replace " wrote Waverley," it is plain that any statement apparently about ( requires (1) ( and (2) ; here (1) states that at least one object satisfies , while (2) states that at most one object satisfies . The two together are equivalent to Thus "" must be part of what is affirmed by any proposition about (. If our proposition is , what is further affirmed is , if . Thus we have i.e. "the satisfying satisfies " is to mean: "There is an object such that is true when, and only when, is , and is true," or, more exactly: "There is a such that '' is always equivalent to ' is ,' and ."

The proposition "" is easily shown to be equivalent to "." For, by the definition, it is i.e. "there is a for which , and this is ," which is equivalent to "." Thus "Scott is the author of Waverley" is equivalent to: i.e. " wrote Waverley" is true when is Scott and false when is not Scott.

When , as above defined, forms part of some other proposition, we shall say that ( has a secondary occurrence. When ( has a secondary occurrence, a proposition in which it occurs may be true even when ( does not exist. This applies, e.g. to the proposition: "There is no such person as the King of France." We may interpret this as if "" stands for " is King of France." In either case, what is asserted is that a proposition in which ( occurs is false, and this proposition is thus part of a larger proposition. The same applies to such a proposition as the following: "If France were a monarchy, the King of France would be of the House of Orleans."

It should be observed that such a proposition as is ambiguous; it may deny , in which case it will be true if ( does not exist, or it may mean in which case it can only be true if ( exists. In ordinary language, the latter interpretation would usually be adopted. For example, the proposition "the King of France is not bald" would usually be rejected as false, being held to mean "the King of France exists and is not bald," rather than "it is false that the King of France exists and is bald." When ( exists, the two interpretations of the ambiguity give equivalent results; but when ( does not exist, one interpretation is true and one is false. It is necessary to be able to distinguish these in our notation; and generally, if we have such propositions as and so on, we must be able by our notation to distinguish whether the whole or only part of the proposition concerned is to be treated as the "" of our definition. For this purpose, we will put "[(]" followed by dots at the beginning of the part (or whole) which is to be taken as , the dots being sufficiently numerous to bracket off the ; i.e. is to be everything following the dots until we reach an equal number of dots not signifying a logical product, or a greater number signifying a logical product, or the end of the sentence, or the end of a bracket enclosing "[(]." Thus It is important to distinguish these two, for if ( does not exist, the first is true and the second false. Again Here again, when ( does not exist, the first is false and the second true.

In order to avoid this ambiguity in propositions containing (, we amend our definition, or rather our notation, putting By means of this definition, we avoid any doubt as to the portion of our whole asserted proposition which is to be treated as the "" of the definition. This portion will be called the scope of (. Thus in the scope of ( is ; but in the scope is ; but in

It will be seen that when ( has the whole of the proposition concerned for its scope, the proposition concerned cannot be true unless ; but when ( has only part of the proposition concerned for its scope, it may often be true even when ( does not exist. It will be seen further that when , we may enlarge or diminish the scope of ( as much as we please without altering the truth-value of any proposition in which it occurs.

It will be found that, in most cases in which descriptions occur, their scope is, in practice, the smallest proposition enclosed in dots or other brackets in which they are contained. Thus for example will occur much more frequently than For this reason it is convenient to decide that, when the scope of an occurrence of ( is the smallest proposition, enclosed in dots or other brackets, in which the occurrence in question is contained, the scope need not be indicated by "[(]." Thus e.g.

This convention enables us, in the vast majority of cases that actually occur, to dispense with the explicit indication of the scope of a descriptive symbol; and it will be found that the convention agrees very closely with the tacit conventions of ordinary language on this subject. Thus for example, if "(" is "the so-and-so," "" is to be read " is not the so-and-so," which would ordinarily be regarded as implying that "the so-and-so" exists; but "" is to be read "it is not true that is the so-and-so," which would generally be allowed to hold if "the so-and-so" does not exist. Ordinary language is, of course, rather loose and fluctuating in its implications on this matter; but subject to the requirement of definiteness, our convention seems to keep as near to ordinary language as possible.

In the case when the smallest proposition enclosed in dots or other brackets contains two or more descriptions, we shall assume, in the absence of any indication to the contrary, that one which typographically occurs earlier has a larger scope than one which typographically occurs later. Thus

(2) Classes. The symbols for classes, like those for descriptions, are, in our system, incomplete symbols: their uses are defined, but they themselves are not assumed to mean anything at all. That is to say, the uses of such symbols are so defined that, when the definiens is substituted for the definiendum, there no longer remains any symbol which could be supposed to represent a class. Thus classes, so far as we introduce them, are merely symbolic or linguistic conveniences, not genuine objects as their members are if they are individuals.

It is an old dispute whether formal logic should concern itself mainly with intensions or with extensions. In general, logicians whose training was mainly philosophical have decided for intensions, while those whose training was mainly mathematical have decided for extensions. The facts seem to be that, while mathematical logic requires extensions, philosophical logic refuses to supply anything except intensions. Our theory of classes recognizes and reconciles these two apparently opposite facts, by showing that an extension (which is the same as a class) is an incomplete symbol, whose use always acquires its meaning through a reference to intension.

In the case of descriptions, it was possible to prove that they are incomplete symbols. In the case of classes, we do not know of any equally definite proof, though arguments of more or less cogency can be elicited from the ancient problem of the One and the Many[31]. It is not necessary for our purposes, however, to assert dogmatically that there are no such things as classes. It is only necessary for us to show that the incomplete symbols which we introduce as representatives of classes yield all the propositions for the sake of which classes might be thought essential. When this has been shown, the mere principle of economy of primitive ideas leads to the non-introduction of classes except as incomplete symbols.

Two propositional functions are said to be formally equivalent when they are equivalent with every possible argument, i.e. when any argument which satisfies the one satisfies the other, and vice versa. Thus " is a man" is formally equivalent to " is a featherless biped"; " is an even prime" is formally equivalent to " is identical with 2."

A function of a function is called extensional when its truth-value with any argument is the same as with any formally equivalent argument. That is to say, is an extensional function of if, provided is formally equivalent to , is equivalent to . Here the apparent variables and are necessarily of the type from which arguments can significantly be supplied to . We find no need to use as apparent variables any functions of non-predicative types; accordingly in the sequel all extensional functions considered are in fact functions of predicative functions[32].

The nature and importance of the distinction between intensional and extensional functions will be made clearer by some illustrations. The proposition "' is a man' always implies ' is a mortal'" is an extensional function of the function " is a man," because we may substitute, for " is a man," " is a featherless biped," or any other statement which applies to the same objects to which " is a man" applies, and to no others. But the proposition " believes that ' is a man' always implies ' is a mortal'" is an intensional function of " is a man," because may never have considered the question whether featherless bipeds are mortal, or may believe wrongly that there are featherless bipeds which are not mortal. Thus even if " is a featherless biped" is formally equivalent to " is a man," it by no means follows that a person who believes that all men are mortal must believe that all featherless bipeds are mortal, since he may have never thought about featherless bipeds, or have supposed that featherless bipeds were not always men. Again the proposition "the number of arguments that satisfy the function is " is an extensional function of , because its truth or falsehood is unchanged if we substitute for any other function which is true whenever is true, and false whenever is false. But the proposition " asserts that the number of arguments satisfying is n" is an intensional function of , since, if asserts this concerning , he certainly cannot assert it concerning all predicative functions that are equivalent to , because life is too short.

Again, consider the proposition "two white men claim to have reached the North Pole." This proposition states "two arguments satisfy the function ' is a white man who claims to have reached the North Pole.'" The truth or falsehood of this proposition is unaffected if we substitute for " is a white man who claims to have reached the North Pole" any other statement which holds of the same arguments, and of no others. Hence it is an extensional function. But the proposition "it is a strange coincidence that two white men should claim to have reached the North Pole," which states "it is a strange coincidence that two arguments should satisfy the function ' is a white man who claims to have reached the North Pole,'" is not equivalent to "it is a strange coincidence that two arguments should satisfy the function ' is Dr Cook or Commander Peary.'" Thus "it is a strange coincidence that should be satisfied by two arguments" is an intensional function of .

The above instances illustrate the fact that the functions of functions with which mathematics is specially concerned are extensional, and that intensional functions of functions only occur where non-mathematical ideas are introduced, such as what somebody believes or affirms, or the emotions aroused by some fact. Hence it is natural, in a mathematical logic, to lay special stress on extensional functions of functions.

When two functions are formally equivalent, we may say that they have the same extension. In this definition, we are in close agreement with usage. We do not assume that there is such a thing as an extension: we merely define the whole phrase "having the same extension." We may now say that an extensional function of a function is one whose truth or falsehood depends only upon the extension of its argument. In such a case, it is convenient to regard the statement concerned as being about the extension. Since extensional functions are many and important, it is natural to regard the extension as an object, called a class, which is supposed to be the subject of all the equivalent statements about various formally equivalent functions. Thus e.g. if we say "there were twelve Apostles," it is natural to regard this statement as attributing the property of being twelve to a certain collection of men, namely those who were Apostles, rather than as attributing the property of being satisfied by twelve arguments to the function " was an Apostle." This view is encouraged by the feeling that there is something which is identical in the case of two functions which "have the same extension." And if we take such simple problems as "how many combinations can be made of things?" it seems at first sight necessary that each "combination" should be a single object which can be counted as one. This, however, is certainly not necessary technically, and we see no reason to suppose that it is true philosophically. The technical procedure by which the apparent difficulty is overcome is as follows.

We have seen that an extensional function of a function may be regarded as a function of the class determined by the argument-function, but that an intensional function cannot be so regarded. In order to obviate the necessity of giving different treatment to intensional and extensional functions of functions, we construct an extensional function derived from any function of a predicative function , and having the property of being equivalent to the function from which it is derived, provided this function is extensional, as well as the property of being significant (by the help of the systematic ambiguity of equivalence) with any argument whose arguments are of the same type as those of . The derived function, written "," is defined as follows: Given a function ), our derived function is to be "there is a predicative function which is formally equivalent to and satisfies ." If is a predicative function, our derived function will be true whenever ) is true. If ) is an extensional function, and is a predicative function, our derived function will not be true unless ) is true; thus in this case, our derived function is equivalent to ). If ) is not an extensional function, and if is a predicative function, our derived function may sometimes be true when the original function is false. But in any case the derived function is always extensional.

In order that the derived function should be significant for any function , of whatever order, provided it takes arguments of the right type, it is necessary and sufficient that ) should be significant, where is any predicative function. The reason of this is that we only require, concerning an argument , the hypothesis that it is formally equivalent to some predicative function , and formal equivalence has the same kind of systematic ambiguity as to type that belongs to truth and falsehood, and can therefore hold between functions of any two different orders, provided the functions take arguments of the same type. Thus by means of our derived function we have not merely provided extensional functions everywhere in place of intensional functions, but we have practically removed the necessity for considering differences of type among functions whose arguments are of the same type. This effects the same kind of simplification in our hierarchy as would result from never considering any but predicative functions.

If ) can be built up by means of the primitive ideas of disjunction, negation, (, and (, as is the case with all the functions of functions that explicitly occur in the present work, it will be found that, in virtue of the systematic ambiguity of the above primitive ideas, any function whose arguments are of the same type as those of can significantly be substituted for in f without any other symbolic change. Thus in such a case what is symbolically, though not really, the same function can receive as arguments functions of various different types. If, with a given argument , the function ), so interpreted, is equivalent to ) whenever is formally equivalent to , then is equivalent to ) provided there is any predicative function formally equivalent to . At this point, we make use of the axiom of reducibility, according to which there always is a predicative function formally equivalent to .

As was explained above, it is convenient to regard an extensional function of a function as having for its argument not the function, but the class determined by the function. Now we have seen that our derived function is always extensional. Hence if our original function was ), we write the derived function , where ")" may be read "the class of arguments which satisfy ," or more simply "the class determined by ." Thus "" will mean: "There is a predicative function which is formally equivalent to and is such that ) is true." This is in reality a function of , but we treat it symbolically as if it had an argument ). By the help of the axiom of reducibility, we find that the usual properties of classes result. For example, two formally equivalent functions determine the same class, and conversely, two functions which determine the same class are formally equivalent. Also to say that is a member of ), i.e. of the class determined by , is true when is true, and false when is false. Thus all the mathematical purposes for which classes might seem to be required are fulfilled by the purely symbolic objects ) provided we assume the axiom of reducibility.

In virtue of the axiom of reducibility, if is any function, there is a formally equivalent predicative function ; then the class ) is identical with the class ), so that every class can be defined by a predicative function. Hence the totality of the classes to which a given term can be significantly said to belong or not to belong is a legitimate totality, although the totality of functions which a given term can be significantly said to satisfy or not to satisfy is not a legitimate totality. The classes to which a given term belongs or does not belong are the classes defined by -functions; they are also the classes defined by predicative -functions. Let us call them -classes. Then "-classes" form a legitimate totality, derived from that of predicative -functions. Hence many kinds of general statements become possible which would otherwise involve vicious-circle paradoxes. These general statements are none of them such as lead to contradictions, and many of them such as it is very hard to suppose illegitimate. The fact that they are rendered possible by the axiom of reducibility, and that they would otherwise be excluded by the vicious-circle principle, is to be regarded as an argument in favour of the axiom of reducibility.

The above definition of "the class defined by the function ," or rather, of any proposition in which this phrase occurs, is, in symbols, as follows: In order to recommend this definition, we shall enumerate five requisites which a definition of classes must satisfy, and we shall then show that the above definition satisfies these five requisites.