# Russell & Whitehead's Principia Mathematica/Introduction/Chapter 3

$\scriptstyle {\frac {d}{dx}}$
$\scriptstyle {\int _{a}^{b}}$
$\scriptstyle {\nabla ^{2}={\frac {\partial ^{2}}{\partial x^{2}}}+{\frac {\partial ^{2}}{\partial y^{2}}}+{\frac {\partial ^{2}}{\partial z^{2}}}\quad {\text{Df}}}$,
$\scriptstyle {\nabla ^{2}}$
$\scriptstyle {\nabla ^{2}}$
$\scriptstyle {x}$
"$\scriptstyle {\sim \mathbf {E!}$℩$\scriptstyle {x)(\phi x)}$,"
$\scriptstyle {\sim \{(\exists c):\phi x.\equiv _{x}.x=c\}}$,
$\scriptstyle {$
$\scriptstyle {x)(\phi x)}$
$\scriptstyle {\sim \mathbf {E!}$
$\scriptstyle {x)(\phi x)}$
$\scriptstyle {$
$\scriptstyle {x)(\phi x)}$
$\scriptstyle {$
$\scriptstyle {x)(\phi x)}$
$\scriptstyle {$
$\scriptstyle {x)(x}$
$\scriptstyle}$
$\scriptstyle {c}$
$\scriptstyle {c}$
$\scriptstyle {c}$
$\scriptstyle {c}$
$\scriptstyle {a=$℩$\scriptstyle {x)(\phi x)}$
$\scriptstyle {a=a}$
$\scriptstyle {$
$\scriptstyle {x)(\phi x)}$
$\scriptstyle {a=$
$\scriptstyle {x)(\phi x)}$
$\scriptstyle {a=a}$
$\scriptstyle {a=$
$\scriptstyle {x)(\phi x)}$
$\scriptstyle {a=y}$
$\scriptstyle {$
$\scriptstyle {x)(\phi x)}$
$\scriptstyle {y}$
$\scriptstyle {y}$
$\scriptstyle {$
$\scriptstyle {x)(\phi x)}$
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 everyone 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.
$\scriptstyle {$
$\scriptstyle {x)(\phi x)}$
$\scriptstyle {$
$\scriptstyle {x)(\phi x)}$
$\scriptstyle {\phi x}$
$\scriptstyle {x}$
$\scriptstyle {$
$\scriptstyle {x)(\phi x)}$
$\scriptstyle {(\exists x).(\phi x)}$
$\scriptstyle {\phi x.\phi y.\supset _{x,y}.x=y}$
$\scriptstyle {\phi x}$
$\scriptstyle {\phi x}$
$\scriptstyle {(\exists c):\phi x.\equiv _{x}.x=c}$,
$\scriptstyle {\mathbf {E!}$℩$\scriptstyle {x)(\phi x)}$.
$\scriptstyle {\mathbf {E!}$
$\scriptstyle {x)(\phi x)}$
$\scriptstyle {$
$\scriptstyle {x)(\phi x)}$
$\scriptstyle {f\{$
$\scriptstyle {x)(\phi x)\}$
$\scriptstyle {fc}$
$\scriptstyle {\phi x.\equiv _{x}.x=c}$
$\scriptstyle {f\{$℩$\scriptstyle {x)(\phi x)\}.=:(\exists c):\phi x.\equiv _{x}.x=c:fc\quad {\text{Df}}$,
$\scriptstyle {x}$
$\scriptstyle {\phi x}$
$\scriptstyle {fx}$
$\scriptstyle {c}$
$\scriptstyle {\phi x}$
$\scriptstyle {x}$
$\scriptstyle {c}$
$\scriptstyle {fc}$
$\scriptstyle {c}$
$\scriptstyle {\phi x}$
$\scriptstyle {x}$
$\scriptstyle {c}$
$\scriptstyle {fc}$
$\scriptstyle {$
$\scriptstyle {x)(\phi x)}$
$\scriptstyle {$
$\scriptstyle {x)(\phi x)}$
$\scriptstyle {a=$
$\scriptstyle {x)(\phi x)}$
$\scriptstyle {\phi x.\equiv _{x}.x=a}$
$\scriptstyle {(\exists c):\phi x.\equiv _{x}.x=c:a=c}$,
$\scriptstyle {c}$
$\scriptstyle {\phi x.\equiv _{x}.x=c}$
$\scriptstyle {c}$
$\scriptstyle {a}$
$\scriptstyle {\phi x.\equiv _{x}.x=a}$
"'$\scriptstyle {x}$ wrote Waverley' is always equivalent to '$\scriptstyle {x}$ is Scott,'"
$\scriptstyle {x}$
$\scriptstyle {x}$
$\scriptstyle {x}$
Thus although "$\scriptstyle {$℩$\scriptstyle {x)(\phi x)}$" has no meaning by itself, it may be substituted for $\scriptstyle {y}$ in any propositional function $\scriptstyle {fy}$, and we get a significant proposition, though not a value of $\scriptstyle {fy}$.
$\scriptstyle {f\{$
$\scriptstyle {x)(\phi x)\}$
$\scriptstyle {$
$\scriptstyle {x)(\phi x)}$
$\scriptstyle {$
$\scriptstyle {x)(\phi x)}$
$\scriptstyle {$
$\scriptstyle {x)(\phi x)}$
$\scriptstyle {\sim \{\mathbf {E!}$℩$\scriptstyle {x)(\phi x)\}$,
$\scriptstyle {\sim \{(\exists c).c=$℩$\scriptstyle {x)(\phi x)\}$,
$\scriptstyle {\phi x}$
$\scriptstyle {x}$
$\scriptstyle {p}$
$\scriptstyle {$
$\scriptstyle {x)(\phi x)}$
$\scriptstyle {p}$
$\scriptstyle {\sim f\{$℩$\scriptstyle {x)(\phi x)\}$
$\scriptstyle {f\{$
$\scriptstyle {x)(\phi x)\}$
$\scriptstyle {$
$\scriptstyle {x)(\phi x)}$
$\scriptstyle {(\exists c):\phi x.\equiv _{x}.x=c:\sim fc}$,
$\scriptstyle {$
$\scriptstyle {x)(\phi x)}$
$\scriptstyle {$
$\scriptstyle {x)(\phi x)}$
$\scriptstyle {$
$\scriptstyle {x)(\phi x)}$
$\scriptstyle {\psi$℩$\scriptstyle {x)(\phi x).\supset .p}$, $\scriptstyle {p.\supset .\psi$℩$\scriptstyle {x)(\phi x)}$, $\scriptstyle {\psi$℩$\scriptstyle {x)(\phi x).\supset .\chi$℩$\scriptstyle {x)(\phi x)}$,
$\scriptstyle {f$
$\scriptstyle {x)(\phi x)}$
$\scriptstyle {[$
$\scriptstyle {x)(\phi x)]}$
$\scriptstyle {f$
$\scriptstyle {x)(\phi x)}$
$\scriptstyle {f$
$\scriptstyle {x)(\phi x)}$
$\scriptstyle {f$
$\scriptstyle {x)(\phi x)}$
$\scriptstyle {[$
$\scriptstyle {x)(\phi x)]}$
$\scriptstyle {[$℩$\scriptstyle {x)(\phi x)].\psi$℩$\scriptstyle {x)(\phi x).\supset .p}$
$\scriptstyle {(\exists c):\phi x.\equiv _{x}.x=c:\psi c:\supset .p}$,
$\scriptstyle {[$℩$\scriptstyle {x)(\phi x)]:\psi$℩$\scriptstyle {x)(\phi x)(\phi x).\supset .p}$
$\scriptstyle {(\exists c):\phi x.\equiv _{x}.x=c:\psi c.\supset .p}$.
$\scriptstyle {$
$\scriptstyle {x)(\phi x)}$
$\scriptstyle {[$℩$\scriptstyle {x)(\phi x)].\sim \psi$℩$\scriptstyle {x)(\phi x)}$
$\scriptstyle {(\exists c):\phi x.\equiv _{x}.x=c:\sim \psi c}$
$\scriptstyle {\sim \{[$℩$\scriptstyle {x)(\phi x)].\sim \psi$℩$\scriptstyle {x)(\phi x)\}$
$\scriptstyle {\sim \{(\exists c):\phi x.\equiv _{x}.x=c:\psi c\}}$.
$\scriptstyle {$
$\scriptstyle {x)(\phi x)}$
$\scriptstyle {$
$\scriptstyle {x)(\phi x)}$
$\scriptstyle {[$℩$\scriptstyle {x)(\phi x)].f$℩$\scriptstyle {x)(\phi x).=:(\exists c):\phi x.\equiv _{x}.x=c:fc\quad {\text{Df.}}}$
$\scriptstyle {f$
$\scriptstyle {x)(\phi x)}$
$\scriptstyle {f$
$\scriptstyle {x)(\phi x)}$
$\scriptstyle {[$℩$\scriptstyle {x)(\phi x)].f$℩$\scriptstyle {x)(\phi x).\supset .p}$
$\scriptstyle {$
$\scriptstyle {x)(\phi x)}$
$\scriptstyle {f$
$\scriptstyle {x)(\phi x)}$
$\scriptstyle {[$℩$\scriptstyle {x)(\phi x)]:f$℩$\scriptstyle {x)(\phi x).\supset .p}$
$\scriptstyle {f$℩$\scriptstyle {x)(\phi x).\supset .p}$;
$\scriptstyle {\sim \{[$℩$\scriptstyle {x)(\phi x)].f$℩$\scriptstyle {x)(\phi x)\}$
$\scriptstyle {f$
$\scriptstyle {x)(\phi x)}$
$\scriptstyle {[$℩$\scriptstyle {x)(\phi x)].\sim f$℩$\scriptstyle {x)(\phi x)\}$
$\scriptstyle {\sim f$℩$\scriptstyle {x)(\phi x)}$.
It will be seen that when $\scriptstyle {$℩$\scriptstyle {x)(\phi x)}$ has the whole of the proposition concerned for its scope, the proposition concerned cannot be true unless $\scriptstyle {\mathbf {E!}$℩$\scriptstyle {x)(\phi x)}$; but when $\scriptstyle {$℩$\scriptstyle {x)(\phi x)}$ has only part of the proposition concerned for its scope, it may often be true even when $\scriptstyle {$℩$\scriptstyle {x)(\phi x)}$ does not exist. It will be seen further that when $\scriptstyle {\mathbf {E!}$℩$\scriptstyle {x)(\phi x)}$, we may enlarge or diminish the scope of $\scriptstyle {$℩$\scriptstyle {x)(\phi x)}$ as much as we please without altering the truth-value of any proposition in which it occurs.
$\scriptstyle {$
$\scriptstyle {x)(\phi x)}$
$\scriptstyle {$
$\scriptstyle {x)(\psi x)}$
$\scriptstyle {[$℩$\scriptstyle {x)(\phi x)]:[$℩$\scriptstyle {x)(\psi x)].f\{$℩$\scriptstyle {x)(\phi x),$℩$\scriptstyle {x)(\psi x)\}$,
$\scriptstyle {[$℩$\scriptstyle {x)(\psi x)]:[$℩$\scriptstyle {x)(\phi x)].f\{$℩$\scriptstyle {x)(\phi x),$℩$\scriptstyle {x)(\psi x)\}$,
$\scriptstyle {$
$\scriptstyle {x)(\phi x)}$
$\scriptstyle {(\exists c):\phi x.\equiv _{x}.x=c:[$℩$\scriptstyle {x)(\psi x)].f\{c,$℩$\scriptstyle {x)(\psi x)\}$,
$\scriptstyle {$
$\scriptstyle {x)(\psi x)}$
$\scriptstyle {(\exists c):.\phi x.\equiv _{x}.x=c:.(\exists d):\psi x.\equiv _{x}.x=c:f(c,d)}$,
$\scriptstyle {$
$\scriptstyle {x)(\psi x)}$
$\scriptstyle {$
$\scriptstyle {x)(\phi x)}$
$\scriptstyle {$
$\scriptstyle {x)(\phi x)}$
$\scriptstyle {$
$\scriptstyle {x)(\psi x)}$
$\scriptstyle {\exists d):.\psi x.\equiv _{x}.x=d:.(\exists c):\phi x.\equiv _{x}.x=c:f(c,d)}$.
(4) and (5) are equivalent, so that the truth-value of a proposition containing two descriptions is independent of the question which has the larger scope.
$\scriptstyle {[$℩$\scriptstyle {x)(\phi x)].\psi$℩$\scriptstyle {x)(\phi x).\supset .[$℩$\scriptstyle {x)(\phi x)].\chi$℩$\scriptstyle {x)(\phi x)}$
$\scriptstyle {[$℩$\scriptstyle {x)(\phi x)]:\psi$℩$\scriptstyle {x)(\phi x).\supset .\chi$℩$\scriptstyle {x)(\phi x)}$.
$\scriptstyle {$
$\scriptstyle {x)(\phi x)}$
$\scriptstyle {[$
$\scriptstyle {x)(\phi x)]}$
$\scriptstyle {p.\supset .a=$℩$\scriptstyle {x)(\phi x)}$
$\scriptstyle {p.\supset .[$℩$\scriptstyle {x)(\phi x)].a=$℩$\scriptstyle {x)(\phi x)}$;
$\scriptstyle {p.\supset .(\exists a).a=$℩$\scriptstyle {x)(\phi x)}$
$\scriptstyle {p.\supset .(\exists a).[$℩$\scriptstyle {x)(\phi x)].a=$℩$\scriptstyle {x)(\phi x)}$;
$\scriptstyle {p.\supset .a\neq$℩$\scriptstyle {x)(\phi x)}$
$\scriptstyle {p.\supset .[$℩$\scriptstyle {x)(\phi x)].\sim \{a=$℩$\scriptstyle {x)(\phi x)\}$;
$\scriptstyle {p.\supset .\sim \{a=$℩$\scriptstyle {x)(\phi x)\}$
$\scriptstyle {p.\supset .\sim \{[$℩$\scriptstyle {x)(\phi x)].a=$℩$\scriptstyle {x)(\phi x)\}$.
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 "$\scriptstyle {$℩$\scriptstyle {x)(\phi x)}$" is "the so-and-so," "$\scriptstyle {a\neq$℩$\scriptstyle {x)(\phi x)}$" is to be read "$\scriptstyle {a}$ is not the so-and-so," which would ordinarily be regarded as implying that "the so-and-so" exists; but "$\scriptstyle {\sim \{a=$℩$\scriptstyle {x)(\phi x)\}$" is to be read "it is not true that $\scriptstyle {a}$ 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.
$\scriptstyle {$℩$\scriptstyle {x)(\phi x)=$℩$\scriptstyle {x)(\psi x)}$
$\scriptstyle {(\exists c):\phi x.\equiv _{x}.x=c:[$℩$\scriptstyle {x)(\psi x)].c=$℩$\scriptstyle {x)(\psi x)}$,
$\scriptstyle {$℩$\scriptstyle {x)(\psi x)=$℩$\scriptstyle {x)(\phi x)}$
$\scriptstyle {(\exists d):\psi x.\equiv _{x}.x=d:[$℩$\scriptstyle {x)(\phi x)].$℩$\scriptstyle {x)(\phi x)=d}$.
These two propositions are easily shown to be equivalent.
(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. 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.
​To explain the theory of classes, it is necessary first to explain the distinction between extensional and intensional functions. This is effected by the following definitions:
The truth-value of a proposition is truth if it is true, and falsehood if it is false. (This expression is due to Frege.)
Two propositions are said to be equivalent when they have the same truth-value, i.e. when they are both true or both false.
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 "$\scriptstyle {\hat {x}}$ is a man" is formally equivalent to "$\scriptstyle {\hat {x}}$ is a featherless biped"; "$\scriptstyle {\hat {x}}$ is an even prime" is formally equivalent to "$\scriptstyle {\hat {x}}$ 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, $\scriptstyle {f(\phi {\hat {z}})}$ is an extensional function of $\scriptstyle {\phi {\hat {z}}}$ if, provided $\scriptstyle {\psi {\hat {z}}}$ is formally equivalent to $\scriptstyle {\phi {\hat {z}}}$, $\scriptstyle {f(\phi {\hat {z}})}$ is equivalent to $\scriptstyle {f(\psi {\hat {z}})}$. Here the apparent variables $\scriptstyle {\phi }$ and $\scriptstyle {\psi }$ are necessarily of the type from which arguments can significantly be supplied to $\scriptstyle {f}$. 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.
A function of a function is called intensional when it is not extensional.
The nature and importance of the distinction between intensional and extensional functions will be made clearer by some illustrations. The proposition "'$\scriptstyle {x}$ is a man' always implies '$\scriptstyle {x}$ is a mortal'" is an extensional function of the function "$\scriptstyle {\hat {x}}$ is a man," because we may substitute, for "$\scriptstyle {x}$ is a man," "$\scriptstyle {x}$ is a featherless biped," or any other statement which applies to the same objects to which "$\scriptstyle {x}$ is a man" applies, and to no others. But the proposition "$\scriptstyle {A}$ believes that '$\scriptstyle {x}$ is a man' always implies '$\scriptstyle {x}$ is a mortal'" is an intensional function of "$\scriptstyle {\hat {x}}$ is a man," because $\scriptstyle {A}$ 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 "$\scriptstyle {x}$ is a featherless biped" is formally equivalent to "$\scriptstyle {x}$ 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 $\scriptstyle {\phi !{\hat {z}}}$ is $\scriptstyle {n}$" is an extensional function of $\scriptstyle {\phi !{\hat {z}}}$, because its truth or falsehood is unchanged if we substitute for $\scriptstyle {\phi !{\hat {z}}}$ any other function which is true whenever $\scriptstyle {\phi !{\hat {z}}}$ is true, and false whenever $\scriptstyle {\phi !{\hat {z}}}$ is false. But the proposition "$\scriptstyle {A}$ asserts that the number of arguments satisfying $\scriptstyle {\phi !{\hat {z}}}$ is $\scriptstyle {n}$" is an intensional function of $\scriptstyle {\phi !{\hat {z}}}$, ​since, if $\scriptstyle {A}$ asserts this concerning $\scriptstyle {\phi !{\hat {z}}}$, he certainly cannot assert it concerning all predicative functions that are equivalent to $\scriptstyle {\phi !{\hat {z}}}$, 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 '$\scriptstyle {\hat {x}}$ 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 "$\scriptstyle {\hat {x}}$ 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 '$\scriptstyle {\hat {x}}$ 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 '$\scriptstyle {\hat {x}}$ is Dr Cook or Commander Peary.'" Thus "it is a strange coincidence that $\scriptstyle {\phi !{\hat {x}}}$ should be satisfied by two arguments" is an intensional function of $\scriptstyle {\phi !{\hat {x}}}$.
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 col1ection of men, namely those who were Apostles, rather than as attributing the property of being satisfied by twelve arguments to the function "$\scriptstyle {x}$ 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 $\scriptstyle {n}$ 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 $\scriptstyle {\psi !{\hat {z}}}$, 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 $\scriptstyle {\phi {\hat {z}}}$ whose arguments are of the same type as those of $\scriptstyle {\psi !{\hat {z}}}$. The derived function, written "$\scriptstyle {f\{{\hat {z}}(\phi z)\}}$," is defined as follows: Given a function $\scriptstyle {f(\psi !{\hat {z}})}$, our derived function is to be "there is a predicative function which is formally equivalent to $\scriptstyle {\phi {\hat {z}}}$ and satisfies $\scriptstyle {f}$." If $\scriptstyle {\phi {\hat {z}}}$ is a predicative function, our derived function will be true whenever $\scriptstyle {f(\phi {\hat {z}})}$ is true. If $\scriptstyle {f(\phi {\hat {z}})}$ is an extensional function, and $\scriptstyle {\phi {\hat {z}}}$ is a predicative function, our derived function will not be true unless $\scriptstyle {f(\phi {\hat {z}})}$ is true; thus in this case, our derived function is equivalent to $\scriptstyle {f(\phi {\hat {z}})}$. If $\scriptstyle {f(\phi {\hat {z}}}$ is not an extensional function, and if $\scriptstyle {\phi {\hat {z}}}$ 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 $\scriptstyle {\phi {\hat {z}}}$, of whatever order, provided it takes arguments of the right type, it is necessary and sufficient that $\scriptstyle {f(\psi !{\hat {z}})}$) should be significant, where $\scriptstyle {\psi !{\hat {z}}}$ is any predicative function. The reason of this is that we only require, concerning an argument $\scriptstyle {\phi {\hat {z}}}$, the hypothesis that it is formally equivalent to some predicative function, $\scriptstyle {\psi !{\hat {z}}}$, 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 $\scriptstyle {f(\psi !{\hat {z}})}$ can be built up by means of the primitive ideas of disjunction, negation, $\scriptstyle {(x).\phi x}$, and $\scriptstyle {(\exists x).\phi x}$, 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 $\scriptstyle {\phi {\hat {z}}}$ whose arguments are of the same type as those of $\scriptstyle {\psi !{\hat {z}}}$ can significantly be substituted for $\scriptstyle {\psi !{\hat {z}}}$ in $\scriptstyle {f}$ without any other symbolic change. Thus in ​such a case what is symbolically, though not really, the same function $\scriptstyle {f}$ can receive as arguments functions of various different types. If, with a given argument $\scriptstyle {\phi {\hat {z}}}$, the function $\scriptstyle {f(\phi {\hat {z}})}$, so interpreted, is equivalent to $\scriptstyle {f(\psi !{\hat {z}})}$ whenever $\scriptstyle {\psi !{\hat {z}}}$ is formally equivalent to $\scriptstyle {\phi {\hat {z}}}$, then $\scriptstyle {f\{{\hat {z}}(\phi z)\}}$ is equivalent to $\scriptstyle {f(\phi {\hat {z}})}$ provided there is any predicative function formally equivalent to $\scriptstyle {\phi {\hat {z}}}$. At this point, we make use of the axiom of reducibility, according to which there always is a predicative function formally equivalent to $\scriptstyle {\phi {\hat {z}}}$.
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 $\scriptstyle {f(\psi !{\hat {z}})}$, we write the derived function $\scriptstyle {f\{{\hat {z}}(\phi z)\}}$, where "$\scriptstyle {{\hat {z}}(\phi z)}$" may be read "the class of arguments which satisfy $\scriptstyle {\phi {\hat {z}}}$," or more simply "the class determined by $\scriptstyle {\phi {\hat {z}}}$." Thus "$\scriptstyle {f\{{\hat {z}}(\phi z)\}}$" will mean: "There is a predicative function $\scriptstyle {\psi !{\hat {z}}}$ which is formally equivalent to $\scriptstyle {\phi {\hat {z}}}$ and is such that $\scriptstyle {f(\psi !{\hat {z}})}$ is true." This is in reality a function of $\scriptstyle {\phi {\hat {z}}}$, but we treat it symbolically as if it had an argument $\scriptstyle {{\hat {z}}(\phi z)}$. 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 $\scriptstyle {x}$ is a member of $\scriptstyle {{\hat {z}}(\phi z)}$, i.e. of the class determined by $\scriptstyle {\phi {\hat {z}}}$, is true when $\scriptstyle {\phi x}$ is true, and false when $\scriptstyle {\phi x}$ is false. Thus all the mathematical purposes for which classes might seem to be required are fulfilled by the purely symbolic objects $\scriptstyle {{\hat {z}}(\phi z)}$, provided we assume the axiom of reducibility.
In virtue of the axiom of reducibility, if $\scriptstyle {\phi {\hat {z}}}$ is any function, there is a formally equivalent predicative function $\scriptstyle {\psi !{\hat {z}}}$; then the class $\scriptstyle {{\hat {z}}(\phi z)}$ is identical with the class $\scriptstyle {{\hat {z}}(\psi !z)}$, 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 $\scriptstyle {a}$ belongs or does not belong are the classes defined by $\scriptstyle {a}$-functions; they are also the classes defined by predicative $\scriptstyle {a}$-functions. Let us call them $\scriptstyle {a}$-classes. Then "$\scriptstyle {a}$-classes" form a legitimate totality, derived from that of predicative $\scriptstyle {a}$-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.
$\scriptstyle {\phi {\hat {z}}}$
$\scriptstyle {f\{{\hat {z}}(\phi z)\}.=:(\exists \psi ):\phi x.\equiv _{x}.\psi !x:f\{\psi !{\hat {z}}\}\quad {\text{Df.}}}$
$\scriptstyle {n}$
$\scriptstyle {m}$
$\scriptstyle {n}$
$\scriptstyle {m}$
$\scriptstyle {\alpha \in \alpha }$
$\scriptstyle {\alpha \sim \in \alpha }$
$\scriptstyle {\alpha \sim \in \alpha }$
$\scriptstyle {\kappa }$
$\scriptstyle {\alpha \in \kappa .\equiv _{\alpha }.\alpha \sim \in \alpha }$.
$\scriptstyle {\kappa \in \kappa }$
$\scriptstyle {\alpha }$
$\scriptstyle {\kappa }$
$\scriptstyle {\kappa \in \kappa .\equiv .\kappa \sim \in \kappa }$.
$\scriptstyle {\alpha \in \alpha }$
$\scriptstyle {\alpha \sim \in \alpha }$
$\scriptstyle {\iota 'x=x}$
$\scriptstyle {x\in \iota 'x}$
$\scriptstyle {x=\iota 'x}$
$\scriptstyle {\iota 'x\in \iota 'x}$
$\scriptstyle {x=\iota 'x}$
$\scriptstyle {\text{Cls}}$
$\scriptstyle {{\text{Cls}}\in {\text{Cls}}}$
$\scriptstyle {\text{Cls}}$
$\scriptstyle {{\text{Cls}}\in {\text{Cls}}}$
$\scriptstyle {\text{Cls}}$
As regards the above requisites, it is plain, to begin with, that, in accordance with our definition, every propositional function $\scriptstyle {\phi {\hat {z}}}$ determines a class $\scriptstyle {{\hat {z}}(\phi z)}$. Assuming the axiom of reducibility, there must always be true propositions about $\scriptstyle {{\hat {z}}(\phi z)}$, i.e. true propositions of the form $\scriptstyle {f\{{\hat {z}}(\phi z)\}}$. For suppose $\scriptstyle {\phi {\hat {z}}}$ is formally equivalent to $\scriptstyle {\psi !{\hat {z}}}$, and suppose $\scriptstyle {\psi !{\hat {z}}}$ satisfies some function $\scriptstyle {f}$. Then $\scriptstyle {{\hat {z}}(\phi z)}$ also satisfies $\scriptstyle {f}$. Hence, given any function $\scriptstyle {\phi {\hat {z}}}$, there are true propositions of the form $\scriptstyle {f\{{\hat {z}}(\phi z)\}}$, i.e. true propositions in which "the class determined by $\scriptstyle {\phi {\hat {z}}}$" is grammatically the subject. This shows that our definition fulfils the first of our five requisites.
$\scriptstyle {{\hat {z}}(\phi z)}$
$\scriptstyle {{\hat {z}}(\psi z)}$
$\scriptstyle {{\hat {z}}(\phi z)={\hat {z}}(\psi z).\equiv :\phi x.\equiv _{x}.\psi x}$.
$\scriptstyle {{\hat {z}}(\phi z)={\hat {z}}(\psi z)}$
$\scriptstyle {f\{{\hat {z}}(\phi z)\}}$
"$\scriptstyle {\chi !{\hat {z}}=\theta !{\hat {z}}}$,"
$\scriptstyle {\chi !{\hat {z}}=\theta !{\hat {z}}.=:(f):f!\chi !{\hat {z}}.\supset .f!\theta !{\hat {z}}\quad {\text{Df}}}$
$\scriptstyle {{\hat {z}}(\phi z)={\hat {z}}(\psi z)}$
$\scriptstyle {$
$\scriptstyle {x)(\phi x)}$
$\scriptstyle {$
$\scriptstyle {x)(\psi x)}$
$\scriptstyle {{\hat {z}}(\phi z)={\hat {z}}(\psi z)}$
$\scriptstyle {(\exists \chi ):\phi x.\equiv _{x}.\chi !x:\chi !{\hat {z}}={\hat {z}}(\psi z)}$,
$\scriptstyle {{\hat {z}}(\psi z)}$
$\scriptstyle {(\exists \chi ):.\phi x.\equiv _{x}.\chi !x:.(\exists \theta ):\psi x.\equiv _{x}.\theta !x:\chi !{\hat {z}}=\theta !{\hat {z}}}$,
$\scriptstyle {(\exists \chi ,\theta ):\phi x.\equiv _{x}.\chi !x:\psi x.\equiv _{x}.\theta !x:\chi !{\hat {z}}=\theta !{\hat {z}}}$,
$\scriptstyle {(\exists \chi ):\phi x.\equiv _{x}.\chi !x:\psi x.\equiv _{x}.\chi !x}$,
$\scriptstyle {\phi x.\equiv _{x}.\psi x}$.
$\scriptstyle {{\hat {z}}(\phi z)}$
$\scriptstyle {\vdash :.{\hat {z}}(\phi z)={\hat {z}}(\psi z).\equiv :\phi x.\equiv _{x}.\psi x}$.
$\scriptstyle {x\in {\hat {z}}(\phi z)}$
$\scriptstyle {x}$
$\scriptstyle {\phi {\hat {z}}}$
$\scriptstyle {f\{{\hat {z}}(\phi z)\}}$
$\scriptstyle {f\{\phi !{\hat {z}}\}}$
$\scriptstyle {x\in \psi !{\hat {z}}.=.\psi !x\quad {\text{Df.}}}$
$\scriptstyle {x\in {\hat {z}}(\phi z)}$
$\scriptstyle {f\{{\hat {z}}(\phi z)\}}$
$\scriptstyle {(\exists \psi ):\phi y.\equiv _{y}.\psi !y:\psi !x}$.
$\scriptstyle {x\in {\hat {z}}(\phi z)}$
$\scriptstyle {\phi x}$
$\scriptstyle {\psi !x}$
$\scriptstyle {\psi !x}$
$\scriptstyle {\phi x}$
$\scriptstyle {\phi x}$
$\scriptstyle {x\in {\hat {z}}(\phi z)}$
$\scriptstyle {\psi }$
$\scriptstyle {\phi }$
$\scriptstyle {x}$
$\scriptstyle {\psi }$
$\scriptstyle {x}$
$\scriptstyle {\phi }$
$\scriptstyle {\vdash :x\in {\hat {z}}(\phi z).\equiv .\phi x}$,
$\scriptstyle {x}$
$\scriptstyle {{\hat {z}}(\phi z)}$
$\scriptstyle {x}$
$\scriptstyle {\phi }$
$\scriptstyle {f\{{\hat {z}}(\phi z)\}}$
$\scriptstyle {{\hat {z}}(\phi z)}$
$\scriptstyle {f\{{\hat {z}}(\phi z)\}}$
$\scriptstyle {\alpha }$
$\scriptstyle {{\hat {z}}(\phi z)}$
$\scriptstyle {{\hat {\alpha }}(f\alpha )}$
$\scriptstyle {\alpha }$
$\scriptstyle {f\alpha }$
$\scriptstyle {F\{{\hat {\alpha }}(f\alpha )\}.=:(\exists g):f\beta .\equiv _{\beta }.g!\beta :F\{g!{\hat {\alpha }}\}\quad {\text{Df}}}$,
$\scriptstyle {\beta }$
$\scriptstyle {{\hat {z}}(\psi !z)}$
$\scriptstyle {\gamma \in {\hat {\alpha }}(f\alpha )}$
$\scriptstyle {F\{{\hat {\alpha }}(f\alpha )\}}$
$\scriptstyle {\vdash :.\gamma \in {\hat {\alpha }}(f\alpha ).\equiv :(\exists g):f\beta .\equiv _{\beta }.g!\beta :\gamma \in g!{\hat {\alpha }}}$.
$\scriptstyle {x\in \psi !{\hat {z}}.=.\psi !x\quad {\text{Df}}}$,
$\scriptstyle {\gamma \in g!{\hat {\alpha }}.=.g!\gamma \quad {\text{Df.}}}$
$\scriptstyle {\vdash :.\gamma \in {\hat {\alpha }}(f\alpha ).\equiv :(\exists g):f\beta .\equiv _{\beta }.g!\beta :g!\gamma }$.
$\scriptstyle {(\exists g):f(\psi !{\hat {z}}).\equiv _{\psi }.g!(\psi !{\hat {z}})}$,
$\scriptstyle {\vdash :(\exists g):f\{{\hat {z}}(\psi !z)\}.\equiv _{\psi }.g!\{{\hat {z}}(\psi !z)\}}$,
$\scriptstyle {\vdash :(\exists g):f\beta .\equiv _{\beta }.g!\beta }$.
$\scriptstyle {\vdash :\gamma \in {\hat {\alpha }}(f\alpha ).\equiv .f\gamma }$.
Thus every function which can take classes as arguments, i.e. every function of functions, determines a class of classes, whose members are those classes which satisfy the determining function. Thus the theory of classes of classes offers no difficulty.
$\scriptstyle {{\hat {z}}(\phi z)\in {\hat {z}}(\phi z)}$
$\scriptstyle {f\{{\hat {z}}(\phi z)\}}$
$\scriptstyle {(\exists \psi ):\phi x.\equiv _{x}.\psi !x:\psi !{\hat {z}}\in \psi !{\hat {z}}}$,
$\scriptstyle {x\in \psi !{\hat {z}}.=.\psi !x\quad {\text{Df}}}$,
$\scriptstyle {(\exists \psi ):\phi x.\equiv _{x}.\psi !x:\psi !(\psi !{\hat {z}})}$.
$\scriptstyle {\psi !(\psi !{\hat {z}})}$
$\scriptstyle {{\hat {z}}(\phi z)\in {\hat {z}}(\phi z)}$
$\scriptstyle {f$
$\scriptstyle {x)(\phi x)}$
$\scriptstyle {f\{{\hat {z}}(\phi z)\}}$
$\scriptstyle {{\hat {z}}(\phi z)}$
$\scriptstyle {(\exists \psi ):\phi x.\equiv _{x}.\psi !x}$,
$\scriptstyle {\mathbf {E!}$
$\scriptstyle {x)(\phi x)}$
$\scriptstyle {{\hat {z}}(\phi z)}$
$\scriptstyle {{\hat {z}}(\phi z)}$
$\scriptstyle {{\hat {z}}(\phi z)}$
$\scriptstyle {[{\hat {z}}(\phi z)]}$
$\scriptstyle {[$
$\scriptstyle {x)(\phi x)]}$
Similarly when two class symbols occur, e.g. in a proposition of the form $\scriptstyle {f\{{\hat {z}}(\phi z),{\hat {z}}(\psi z)\}}$, we need not remember rules for the scopes of the two symbols, since all choices give equivalent results, as it is easy to prove. For the preliminary propositions a rule is desirable, so we can decide that the class symbol which occurs first in the order of writing is to have the larger scope.
$\scriptstyle {\alpha }$
$\scriptstyle {\alpha }$
$\scriptstyle {{\hat {z}}(\phi z)}$
$\scriptstyle {{\hat {z}}(\psi z)}$
$\scriptstyle {{\hat {z}}(\chi z)}$
$\scriptstyle {\phi z}$
$\scriptstyle {\psi z}$
$\scriptstyle {\chi z}$
$\scriptstyle {\vdash :\phi x\equiv _{x}\psi x.\supset .f\{{\hat {z}}(\phi z)\}\equiv f\{{\hat {z}}(\psi z)\}}$."
$\scriptstyle {\alpha }$
$\scriptstyle {f(\alpha )}$
$\scriptstyle {\alpha }$
$\scriptstyle {f\{{\hat {z}}(\phi z)\}}$
$\scriptstyle {\phi }$
"$\scriptstyle {(\exists \psi ).\phi x\equiv _{x}\psi !x.f\{\psi !{\hat {z}}\}}$,"
$\scriptstyle {\phi }$
$\scriptstyle {\phi {\hat {z}}}$
$\scriptstyle {\psi {\hat {z}}}$
$\scriptstyle {\alpha }$
$\scriptstyle {fa=.(\exists \psi ).\phi !x\equiv _{x}\psi !x.f\{\psi !{\hat {z}}\}\quad {\text{Df.}}}$
$\scriptstyle {\alpha }$
$\scriptstyle {\phi }$
$\scriptstyle {f{\hat {\alpha }}}$
"$\scriptstyle {(\exists \psi ).{\hat {\phi }}!x\equiv _{x}\psi !x.f\{\psi !{\hat {x}}\}}$,"
$\scriptstyle {(\alpha ).f\alpha }$
"$\scriptstyle {(\phi ):(\exists \psi ).\phi !x\equiv _{x}\psi !x.f\{\psi !{\hat {z}}\}}$."
$\scriptstyle {f!\alpha =.(\exists \psi ).\phi !x\equiv _{x}\psi !x.f!\{\psi !{\hat {z}}\}\quad {\text{Df.}}}$
​(3) Relations. With regard to relations, we have a theory strictly analogous to that which we have just explained as regards classes. Relations in extension, like classes, are incomplete symbols. We require a division of functions of two variables into predicative and non-predicative functions, again for reasons which have been explained in Chapter II. We use the notation "$\scriptstyle {\phi !(x,y)}$" for a predicative function of $\scriptstyle {x}$ and $\scriptstyle {y}$.
$\scriptstyle {\phi !({\hat {x}},{\hat {y}})}$
$\scriptstyle {{\hat {x}}{\hat {y}}\phi (x,y)}$
$\scriptstyle {\phi (x,y)}$
$\scriptstyle {f\{{\hat {x}}{\hat {y}}\phi (x,y)\}.=:(\exists \psi ):\phi (x,y).\equiv _{x,y}.\psi !(x,y):f\{\psi !({\hat {x}},{\hat {y}})\}\quad {\text{Df.}}}$
$\scriptstyle {f\{\psi !({\hat {x}},{\hat {y}})\}}$
$\scriptstyle {\psi }$
$\scriptstyle {f\{{\hat {x}}{\hat {y}}\phi (x,y)\}}$
$\scriptstyle {\phi }$
$\scriptstyle {\vdash :.{\hat {x}}{\hat {y}}\phi (x,y)={\hat {x}}{\hat {y}}\psi (x,y).\equiv :\phi (x,y).\equiv _{x,y}.\psi (x,y)}$,
$\scriptstyle {x\in \psi !{\hat {z}}}$
$\scriptstyle {x\{\psi !({\hat {x}},{\hat {y}})\}y.=.\psi !(x,y)\quad {\text{Df}}}$.
$\scriptstyle {x\in \psi !{\hat {z}}}$
$\scriptstyle {x\{{\hat {x}}{\hat {y}}\phi (x,y)\}y}$.
$\scriptstyle {(\exists \psi ):\phi (x,y).\equiv _{x,y}.\psi !(x,y):x\{\psi !({\hat {x}},{\hat {y}})\}y}$,
$\scriptstyle {(\exists \psi ):\phi (x,y).\equiv _{x,y}.\psi !(x,y):\psi !(x,y)}$,
"$\scriptstyle {(\exists \psi ):\phi (x,y).\equiv _{x,y}.\psi !(x,y)}$,"
$\scriptstyle {\phi (x,y)}$.
$\scriptstyle {\vdash :x\{{\hat {x}}{\hat {y}}\phi (x,y)\}y.\equiv .\phi (x,y)}$.
Whenever the determining function of a relation is not relevant, we may replace $\scriptstyle {{\hat {x}}{\hat {y}}\phi (x,y)}$ by a single capital letter. In virtue of the propositions given above,
| and | ${\begin{aligned}&\scriptstyle {\vdash :.R=S.=:xRy.\equiv _{x,y}.xSy,}\\&\scriptstyle {\vdash :.R={\hat {x}}{\hat {y}}\phi (x,y).\equiv :xRy.\equiv _{x,y}.\phi (x,y),}\\&\scriptstyle {\vdash .R={\hat {x}}{\hat {y}}(xRy).}\end{aligned}}$ |
Classes of relations, and relations of relations, can be dealt with as classes of classes were dealt with above.
​Just as a class must not be capable of being or not being a member of itself, so a relation must not be referent or relatum with respect to itself. This turns out to be equivalent to the assertion that $\scriptstyle {\phi !({\hat {x}},{\hat {y}})}$ cannot significantly be either of the arguments $\scriptstyle {x}$ or $\scriptstyle {y}$ in $\scriptstyle {\phi !(x,y)}$. This principle, again, results from the limitation to the possible arguments to a function explained at the beginning of Chapter II.
We may sum up this whole discussion on incomplete symbols as follows.
$\scriptstyle {$
$\scriptstyle {x)(\phi x)}$
$\scriptstyle {f$
$\scriptstyle {x)(\phi x)}$
$\scriptstyle {f{\hat {z}}}$
$\scriptstyle {\vdash :.\mathbf {E!}$℩$\scriptstyle {x)(\phi x).\supset :(x).fx.\supset .f$℩$\scriptstyle {x)(\phi x)}$, $\scriptstyle {\vdash :$℩$\scriptstyle {x)(\phi x)=$℩$\scriptstyle {x)(\psi x).\supset .f$℩$\scriptstyle {x)(\phi x)\equiv f$℩$\scriptstyle {x)(\psi x)}$, $\scriptstyle {\vdash :\mathbf {E!}$℩$\scriptstyle {x)(\phi x).\supset .$℩$\scriptstyle {x)(\phi x)=$℩$\scriptstyle {x)(\phi x)}$, $\scriptstyle {\vdash :$℩$\scriptstyle {x)(\phi x)=$℩$\scriptstyle {x)(\psi x).\equiv .$℩$\scriptstyle {x)(\psi x)=$℩$\scriptstyle {x)(\phi x)}$, $\scriptstyle {\vdash :$℩$\scriptstyle {x)(\phi x)=$℩$\scriptstyle {x)(\psi x).$℩$\scriptstyle {x)(\psi x)=$℩$\scriptstyle {x)(\chi x).\supset .$℩$\scriptstyle {x)(\phi x)=$℩$\scriptstyle {x)(\chi x)}$.
The use of the symbol "$\scriptstyle {{\hat {x}}(\phi x)}$" (or of a single letter, such as $\scriptstyle {\alpha }$, to represent such a symbol) as if, in "$\scriptstyle {f\{{\hat {x}}(\phi x)\}}$," it directly represented an argument $\scriptstyle {\alpha }$ to a function $\scriptstyle {f{\hat {\alpha }}}$, is rendered possible by the theorems
${\begin{aligned}&\scriptstyle {\vdash :(\alpha ).f\alpha .\supset .f\{{\hat {x}}(\phi x)\},}\\&\scriptstyle {\vdash :{\hat {x}}(\phi x)={\hat {x}}(\psi x).\supset .f\{{\hat {x}}(\phi x)\}\equiv f\{{\hat {x}}(\psi x)\},}\\&\scriptstyle {\vdash .{\hat {x}}(\phi x)={\hat {x}}(\phi x),}\\&\scriptstyle {\vdash :{\hat {x}}(\phi x)={\hat {x}}(\psi x).\equiv .{\hat {x}}(\psi x)={\hat {x}}(\phi x),}\\&\scriptstyle {\vdash :{\hat {x}}(\phi x)={\hat {x}}(\psi x).{\hat {x}}(\psi x)={\hat {x}}(\chi x).\supset .{\hat {x}}(\phi x)={\hat {x}}(\chi x).}\end{aligned}}$
Throughout these propositions the types must be supposed to be properly adjusted, where ambiguity is possible.
The use of the symbol "$\scriptstyle {{\hat {x}}{\hat {y}}\{\phi (x,y)\}}$ (or of a single letter, such as $\scriptstyle {R}$, to represent such a symbol) as if, in "$\scriptstyle {f\{{\hat {x}}{\hat {y}}\phi (x,y)\}}$," it directly represented an argument $\scriptstyle {R}$ to a function $\scriptstyle {f{\hat {R}}}$, is rendered possible by the theorems
${\begin{aligned}&\scriptstyle {\vdash :(R).fR.\supset .f\{{\hat {x}}{\hat {y}}\phi (x,y)\},}\\&\scriptstyle {\vdash :{\hat {x}}{\hat {y}}\phi (x,y)={\hat {x}}{\hat {y}}\phi (x,y)={\hat {x}}{\hat {y}}\psi (x,y).\supset .f\{{\hat {x}}{\hat {y}}\phi (x,y)\}\equiv f\{{\hat {x}}{\hat {y}}\psi (x,y)\},}\\&\scriptstyle {\vdash .{\hat {x}}{\hat {y}}\phi (x,y)={\hat {x}}{\hat {y}}\phi (x,y),}\\&\scriptstyle {\vdash :{\hat {x}}{\hat {y}}\phi (x,y)={\hat {x}}{\hat {y}}\psi (x,y).\equiv .{\hat {x}}{\hat {y}}\psi (x,y)={\hat {x}}{\hat {y}}\phi (x,y),}\\&\scriptstyle {\vdash :{\hat {x}}{\hat {y}}\phi (x,y)={\hat {x}}{\hat {y}}\psi (x,y).{\hat {x}}{\hat {y}}\psi (x,y)={\hat {x}}{\hat {y}}\chi (x,y).}\\&\scriptstyle {\qquad \qquad \qquad \qquad \qquad \quad \supset .{\hat {x}}{\hat {y}}\phi (x,y)={\hat {x}}{\hat {y}}\chi (x,y).}\end{aligned}}$
Throughout these propositions the types must be supposed to be properly adjusted where ambiguity is possible.
​It follows from these three groups of theorems that these incomplete symbols are obedient to the same formal rules of identity as symbols which directly represent objects, so long as we only consider the equivalence of the resulting variable (or constant) values of propositional functions and not their identity. This consideration of the identity of propositions never enters into our formal reasoning.
$\scriptstyle {$
$\scriptstyle {x)(\phi x)}$
$\scriptstyle {(x).fx.\supset .f$℩$\scriptstyle {x)(\phi x)}$,
$\scriptstyle {$
$\scriptstyle {x)(\phi x)}$
$\scriptstyle {f$
$\scriptstyle {x)(\phi x)}$
$\scriptstyle {f{\hat {x}}}$
$\scriptstyle {f{\hat {x}}}$
$\scriptstyle {f$
$\scriptstyle {x)(\phi x)}$
$\scriptstyle {$
$\scriptstyle {x)(\phi x)}$
$\scriptstyle {(x).x=x}$
the round square=the round square.
$\scriptstyle {(x).fx.\supset .f$℩$\scriptstyle {x)(\phi x)}$
$\scriptstyle {\mathbf {E!}$
$\scriptstyle {x)(\phi x)}$
$\scriptstyle {\mathbf {E!}$
$\scriptstyle {x)(\phi x)}$
$\scriptstyle {$
$\scriptstyle {x)(\phi x)}$
$\scriptstyle {\mathbf {E!}$
$\scriptstyle {x)(\phi x)}$
$\scriptstyle {$
$\scriptstyle {x)(\phi x)}$
$\scriptstyle {c=$
$\scriptstyle {x)(\phi x)}$
$\scriptstyle {c=y}$
In regard to classes, the relevance of their incompleteness is somewhat different. It may he illustrated by the fact that we may have
$\scriptstyle {{\hat {z}}(\phi z)=\psi !{\hat {z}}(\phi z)=\chi !{\hat {z}}}$
$\scriptstyle {\psi !{\hat {z}}=\chi !{\hat {z}}}$.
For, by a direct application of the definitions, we find that
$\scriptstyle {\vdash :{\hat {z}}(\phi z)=\psi !{\hat {z}}.\equiv .\phi x\equiv _{x}\psi !x}$.
Thus we shall have
$\scriptstyle {\vdash :\phi x\equiv _{x}\psi !x.\phi x,]phix\equiv _{x}\chi !x.\supset .{\hat {z}}(\phi z)=\psi !{\hat {z}}.{\hat {z}}(\phi z)=\chi !{\hat {z}}}$,
but we shall not necessarily have $\scriptstyle {\psi !{\hat {z}}=\chi !{\hat {z}}}$ under these circumstances, for two functions may well be formally equivalent without being identical; for example,
$\scriptstyle {x=}$Scott$\scriptstyle {.\equiv _{x}.x=}$the author of Waverley,
$\scriptstyle {{\hat {z}}=}$
$\scriptstyle {{\hat {z}}=}$
$\scriptstyle {x=y.x=z.\supset .y=z}$,
$\scriptstyle {x}$
$\scriptstyle {y}$
$\scriptstyle {z}$
$\scriptstyle {{\hat {z}}(\phi z)=\psi !{\hat {z}}}$
$\scriptstyle {x=y}$
$\scriptstyle {\theta !{\hat {z}}=\psi !{\hat {z}}}$
$\scriptstyle {\psi !{\hat {z}}}$
$\scriptstyle {{\hat {z}}(\phi z)}$
$\scriptstyle {{\hat {z}}(\phi z)=\psi !{\hat {z}}.{\hat {z}}(\phi z)=\chi !{\hat {z}}}$.
$\scriptstyle {{\hat {z}}(\phi z)}$
$\scriptstyle {(\exists \theta ):\phi x\equiv _{x}\theta !x.\theta !{\hat {z}}=\psi !{\hat {z}}.\theta !{\hat {z}}=\chi !{\hat {z}}}$,
$\scriptstyle {\psi !{\hat {z}}=\chi !{\hat {z}}}$.
We may say generally that the fact that $\scriptstyle {{\hat {z}}(\phi z)}$ is an incomplete symbol is not relevant so long as we confine ourselves to extensional functions of functions, but is apt to become relevant for other functions of functions.
* ↑ Cf. pp. 31, 32.
* ↑ We shall generally write "$\scriptstyle {f$℩$\scriptstyle {x)(\phi x)}$" rather than "$\scriptstyle {f\{$℩$\scriptstyle {x)(\phi x)\}$" in future.
* ↑ Briefly, these arguments reduce to the following: If there is such an object as a class, it must be in some sense one object. Yet it is only of classes that many can be predicated. Hence, if we admit classes as objects, we must suppose that the same object can be both one and many, which seems impossible.
* ↑ Cf. p. 56.
* ↑ As explained in Chapter I (pp. 25, 26), "$\scriptstyle {x\in \alpha }$" means "$\scriptstyle {x}$ is a member of the class $\scriptstyle {\alpha }$," or, more shortly, "$\scriptstyle {x}$ is an $\scriptstyle {\alpha }$." The definition of this expression in terms of our theory of classes will be given shortly.
* ↑ This is the second of the contradictions discussed at the end of Chapter II.
* ↑ The use of a single letter, such as $\scriptstyle {\alpha }$ or $\scriptstyle {\beta }$, to represent a variable class, will be further explained shortly.
* ↑ This definition raises certain questions as to the two senses of a relation, which are dealt with in *21.
* ↑ Cf. p. 8.