A simple explanation of set theory and Russell’s paradox

A set and a set of sets
The early set theorists, operating in a world of what we now call "naive set theory", loosely defined a set (a class) as a collection of things.
A finite collection of variables, like {x, y, z}, should be a set. An infinite collection of numbers, like the natural numbers N = {1, 2, 3, 4, 5, ...} should also be a set. In geometry, the collection of all points that form a line between two given points is also a set.

A set can also be a collections of sets. The question is can a set contain itself as a member?

If we have a set of all natural numbers, we could certainly have a set of everything that is not a natural number. This set would include quite a few things — the numbers -3, 1/2, and π are all not naturals, and so they would be members. The word "pizza" is not a natural number, so that would be a member. The state of California is not a natural number as well, so we would throw that in there too.

Since this set is itself pretty clearly not a natural number, but instead an enormous collection of everything ever that is not a natural number, it must be a member of itself.

Indeed, with our naive definition of a set, it is tempting to consider a set of everything – “a set of all sets”. Naturally, being itself a set, the set of all sets would also have to contain itself as an element.

Russell's Paradox
Let's first look at the set of all sets that contain themselves as member, and let's call this set A.
We've seen a couple examples of set A — the set of everything that is not a natural number, and the set of all sets.
Does A contain itself?
According to "naive set theory", if A satisfies the condition we set up for being a member of A, we can say that A contains itself as a member. if A doesn't satisfy the condition for being in A, then A is not a member. There is no paradox here.

The paradox comes in when we build the set of all sets that do not contain themselves as members. Let's call this set B.
Does B contain itself?
If we suppose B contains itself as a member. We defined B, however, as the set of all sets that do not contain themselves. So if B does contain itself, it goes against the condition we used to define B, and thus B does not contain itself.
But then if B does not contain itself, it does satisfy the condition to be a member of itself, and so it would have to contain itself.

This gives us a contradiction — the set of all sets that are not members of themselves simultaneously must and cannot be a member of itself. This contradiction makes "naive set theory" inconsistent — we have a statement that has to be simultaneously true and false.

In 1902, Bertrand Russell identified this problem which is known as Russell's Paradox.

Modern Set Theory Axiom
The modern set theory axioms are very specific about how to build sets out of other sets. In particular, the axioms very quickly forbid a set from being a member of itself. We are also much more careful with constructions like "the set of everything that is not a natural number". Rather than using a broad universe of "everything", sets like this must be constructed as subsets of a larger set that we have already defined. So, I can define the set of all real numbers that are not natural numbers, but I cannot make a set of "everything" that is not a natural number.

The axiomatization of set theory

In order to avoid the paradoxes and put it on a firm footing, set theory had to be axiomatized.
The first axiomatization was due to Zermelo (1908). Zermelo's axiomatization avoids Russell's Paradox by means of the Separation axiom, which is formulated as quantifying over properties of sets, and thus it is a second-order statement.
Further work by Skolem and Fraenkel led to the formalization of the Separation axiom in terms of formulas of first-order.
A further addition, by von Neumann, of the axiom of Foundation, led to the standard axiom system of set theory, known as the Zermelo-Fraenkel axioms plus the Axiom of Choice, or ZFC.
Other axiomatizations of set theory, such as those of von Neumann-Bernays-Gödel (NBG), or Morse-Kelley (MK), allow also for a formal treatment of proper classes.

The axioms of set theory
ZFC is an axiom system formulated in first-order logic with equality and with only one binary relation symbol ∈ for membership. Thus, we write A∈B to express that A is a member of the set B. We state below the axioms of ZFC informally.

The axioms of ZFC
Extensionality: If two sets A and B have the same elements, then they are equal.
Null Set: There exists a set, denoted by ∅ and called the empty set, which has no elements.
Pair: Given any sets A and B, there exists a set, denoted by {A,B}, which contains A and B as its only elements. In particular, there exists the set {A} which has A as its only element.
Power Set: For every setA there exists a set, denoted by P(A) and called the power set of A, whose elements are all the subsets of A.
Union: For every set A, there exists a set, denoted by ⋃A and called the union of A, whose elements are all the elements of the elements of A.
Infinity: There exists an infinite set. In particular, there exists a set Z that contains ∅ and such that if A∈Z, then ⋃{A,{A}}∈Z.
Separation: For every set A and every given property, there is a set containing exactly the elements of A that have that property. A property is given by a formula φ of the first-order language of set theory.

Thus, Separation is not a single axiom but an axiom schema, that is, an infinite list of axioms, one for each formula φ.

Replacement: For every given definable function with domain a set A, there is a set whose elements are all the values of the function.
Replacement is also an axiom schema, as definable functions are given by formulas.
Foundation: Every non-empty set A contains an ∈-minimal element, that is, an element such that no element of A belongs to it.

These are the axioms of Zermelo-Fraenkel set theory, or ZF. The axioms of Null Set and Pair follow from the other ZF axioms, so they may be omitted. Also, Replacement implies Separation.

Finally, there is the Axiom of Choice (AC):
Choice: For every set A of pairwise-disjoint non-empty sets, there exists a set that contains exactly one element from each set in A.

The AC was, for a long time, a controversial axiom. On the one hand, it is very useful and of wide use in mathematics. On the other hand, it has rather unintuitive consequences, such as the Banach-Tarski Paradox, which says that the unit ball can be partitioned into finitely-many pieces, which can then be rearranged to form two unit balls. The objections to the axiom arise from the fact that it asserts the existence of sets that cannot be explicitly defined. But Gödel's 1938 proof of its consistency, relative to the consistency of ZF, dispelled any suspicions left about it.

The Axiom of Choice is equivalent, modulo ZF, to the Well-ordering Principle, which asserts that every set can be well-ordered, i.e., it can be linearly ordered so that every non-empty subset has a minimal element.

Although not formally necessary, besides the symbol ∈∈ one normally uses for convenience other auxiliary defined symbols. For example,
A⊆B expresses that A is a subset of B, i.e., every member of A is a member of B.
Other symbols are used to denote sets obtained by performing basic operations, such as
A∪B, which denotes the union of A and B, i.e., the set whose elements are those of A and B; or
A∩B, which denotes the intersection of A and B, i.e., the set whose elements are those common to A and B.
The ordered pair (A,B) is defined as the set {{A},{A,B}}.
Thus, two ordered pairs (A,B) and (C,D) are equal if and only if A=C and B=D.
And the Cartesian product A×B is defined as the set of all ordered pairs (C,D) such that C∈A and D∈B.
Given any formula φ(x,y1,…,yn) , and sets A,B1,…,Bn , one can form the set of all those elements of A that satisfy the formula φ(x, B1,…, Bn). This set is denoted by {a∈A:φ(a,B1,…,Bn)}.
In ZF one can easily prove that all these sets exist.

Russell's Paradox

Naïve set theory assumes the so-called naïve or unrestricted Comprehension Axiom, the axiom that for any formula φ(x) containing x as a free variable, there will exist the set {x : φ(x)} whose members are exactly those objects that satisfy φ(x). Thus, if the formula φ(x) stands for “x is prime”, then {x : φ(x)} will be the set of prime numbers.

But from the assumption of this axiom, Russell’s contradiction follows. For example, if we let φ(x) stand for x ∈ x and let R = {x: ~φ(x)}, then R is the set whose members are exactly those objects that are not members of themselves.

Is R a member of itself? If it is, then it must satisfy the condition of not being a member of itself and so it is not. If it is not, then it must not satisfy the condition of not being a member of itself, and so it must be a member of itself. Since by classical logic one case or the other must hold – either R is a member of itself or it is not – it follows that the theory implies a contradiction.

Russell's Paradox

Russell's paradox (also known as Russell's antinomy), discovered by Bertrand Russell in 1901, showed that the naive set theory of Frege leads to a contradiction.

Consider the set R of all sets that do not contain themselves as members. In set-theoretic notation:
          R = {A | A ∉ A}
Assume, as in Frege's Grundgesetze der Arithmetik, that sets can be freely defined by any condition. Then R is a well-defined set. The problem arises when it is considered whether R is an element of itself. If R is an element of R, then according to the definition, R is not an element of R; if R is not an element of R, then R has to be an element of R, again by its very definition: Hence a contradiction.

Russell's paradox was a primary motivation for the development of set theories with a more elaborate axiomatic basis than simple extensionality and unlimited set abstraction. The paradox drove Russell to develop type theory and Ernst Zermelo to develop an axiomatic set theory, which evolved into the now-canonical Zermelo–Fraenkel set theory.

Russell's Paradox

According to naive set theory, any definable collection is a set.
Let R be the set of all sets that are not members of themselves. If R is not a member of itself, then its definition dictates that it must contain itself, and if it contains itself, then it contradicts its own definition as the set of all sets that are not members of themselves. This contradiction is Russell's paradox. Symbolically:
Let R = {x | x ∉ x}, then R ∈ R ↔ R ∉ R

Set-Theoretic Paradoxes

Russell's paradox arises from considering the Russell set R of all sets that are not members of themselves, that is, the set defined defined by R = {x | x ∉ x }.
The contradiction is then derived by asking whether R is a member of itself, that is, whether R ∈ R holds. If R ∈ R then R is a member of itself, and thus R ∉ R, by definition of R. If, on the other hand, R ∉ R then R is not a member of itself, and thus R ∈ R, again by definition of R.

Russell's Paradox

Russell’s paradox, statement in set theory, devised by the English mathematician-philosopher Bertrand Russell, that demonstrated a flaw in earlier efforts to axiomatize the subject.

Russell found the paradox in 1901 and communicated it in a letter to the German mathematician-logician Gottlob Frege in 1902. Russell’s letter demonstrated an inconsistency in Frege’s axiomatic system of set theory by deriving a paradox within it.

Frege had constructed a logical system employing an unrestricted comprehension principle. The comprehension principle is the statement that, given any condition expressible by a formula ϕ(x), it is possible to form the set of all sets x meeting that condition, denoted {x | ϕ(x)}. For example, the set of all sets—the universal set—would be {x | x = x}.

It was noticed in the early days of set theory, however, that a completely unrestricted comprehension principle led to serious difficulties. In particular, Russell observed that it allowed the formation of {x | x ∉ x}, the set of all non-self-membered sets, by taking ϕ(x) to be the formula x ∉ x. Is this set—call it R—a member of itself? If it is a member of itself, then it must meet the condition of its not being a member of itself. But if it is not a member of itself, then it precisely meets the condition of being a member of itself. This impossible situation is called Russell’s paradox.

The significance of Russell’s paradox is that it demonstrates in a simple and convincing way that one cannot both hold that there is meaningful totality of all sets and also allow an unfettered comprehension principle to construct sets that must then belong to that totality. (Russell spoke of this situation as a “vicious circle.”)

Set theory avoids this paradox by imposing restrictions on the comprehension principle. The standard Zermelo-Fraenkel axiomatization (ZF) does not allow comprehension to form a set larger than previously constructed sets. (The role of constructing larger sets is given to the power-set operation.) This leads to a situation where there is no universal set—an acceptable set must not be as large as the universe of all sets.

A very different way of avoiding Russell’s paradox was proposed in 1937 by the American logician Willard Van Orman Quine. In his paper “New Foundations for Mathematical Logic,” the comprehension principle allows formation of {x | ϕ(x)} only for formulas ϕ(x) that can be written in a certain form that excludes the “vicious circle” leading to the paradox. In this approach, there is a universal set.

Fixing Russell's Paradox

Naive set theory allow mathematicians to construct the set of all sets that satisfy any property. The axiom that says that you're allowed to do this was formally introduced by Frege and is called unrestricted comprehension.

Russell found out that unrestricted comprehension principle makes it possible for mathematicians to construct the set of all sets that don't contain themselves and this leads to a contradiction.

The now-standard fix is to replace unrestricted comprehension with a weaker axiom called restricted comprehension. With restricted comprehension, you're no longer allowed to construct the set of all sets that satisfy some property. Now you're only allowed to construct the set of all elements of some other set that satisfy some property. (In the now-standard approach to set theory, the elements of sets are other sets. It forbids a set from being a member of itself.)

To fix Russell's paradox with restricted comprehension, you need to construct the set of all sets first. And another axiom, foundation, prohibits such a set from existing.

Russell's Paradox and Fixing Russell's Paradox

Russell found the paradox in 1901 and communicated it in a letter to the German mathematician-logician Gottlob Frege in 1902. Russell's letter demonstrated an inconsistency in Frege's axiomatic system of set theory by deriving a paradox within it.

Frege had constructed a logical system employing an unrestricted comprehension principle. The comprehension principle is the statement that, given any condition expressible by a formula ϕ(x), it is possible to form the set of all sets x meeting that condition, denoted {x | ϕ(x)}. For example, the set of all sets—the universal set—would be {x | x = x}.

It was noticed in the early days of set theory, however, that a completely unrestricted comprehension principle led to serious difficulties. In particular, Russell observed that it allowed the formation of {x | x ∉ x}, the set of all non-self-membered sets, by taking ϕ(x) to be the formula x ∉ x. Is this set—call it R—a member of itself? If it is a member of itself, then it must meet the condition of its not being a member of itself. But if it is not a member of itself, then it precisely meets the condition of being a member of itself. This impossible situation is called Russell's paradox.

The significance of Russell's paradox is that it demonstrates in a simple and convincing way that one cannot both hold that there is meaningful totality of all sets and also allow an unfettered comprehension principle to construct sets that must then belong to that totality. (Russell spoke of this situation as a “vicious circle.”)

Set theory avoids this paradox by imposing restrictions on the comprehension principle. The standard Zermelo-Fraenkel axiomatization does not allow comprehension to form a set larger than previously constructed sets. (The role of constructing larger sets is given to the power-set operation.) This leads to a situation where there is no universal set—an acceptable set must not be as large as the universe of all sets.

A very different way of avoiding Russell's paradox was proposed in 1937 by the American logician Willard Van Orman Quine. In his paper “New Foundations for Mathematical Logic,” the comprehension principle allows formation of {x | ϕ(x)} only for formulas ϕ(x) that can be written in a certain form that excludes the “vicious circle” leading to the paradox. In this approach, there is a universal set.

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Fixing Russell's Paradox

When set theory was invented it was assumed that given any predicate, there was a set containing all the things that satisfied the predicate. This assumption is called naive comprehension. Unfortunately this allowed paradoxes like the set of all sets not containing themselves.

So people invented restricted axioms of comprehension. These are rules that say that only certain predicates give rise to sets. There are different kinds of set theory with different comprehension restrictions.

One idea for limiting comprehension is to say that if the class of things satisfies the predicate is too big, the class is not a set. All the commonly used set theories use this idea. The set of all sets is very large indeed, so it is not a set in these theories.

There are other ways of restricting comprehension that get rid of the paradoxical sets but do allow a universal set. One such set theory was invented by Quine and called New Foundations.

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Russell’s paradox arises if you consider the set R = {x : x ∉ x}. Ask yourself if R ∈ R. If you suppose so, then by the definition of unrestricted set comprehension R ∉ R. You have a contradiction, so it must be the opposite of what you supposed, that is, R ∉ R.. But this is the same as saying R belongs to the complement of itself, that is, R ∈ R. You now have another contradiction, but this is far worse, since you have no hypotheses. The whole theory is logically inconsistent.

In set theory there are two ways for getting rid of the Russell’s paradox: either you disallow the set of all sets and other similar sets (see for example the Zermelo-Fraenkel set theory), or you allow them, but you also restrict the way they are used (see for example the Morse-Kelley set theory).

In the first case, set comprehension says if you have a set A you can have {x ∈ A : ϕ(x)} (notice: writing {x : ϕ(x)} is just wrong in this case, because you have to have an initial set). If you now define R = {x ∈ A : x ∉ x} and you repeat the same passages as before, it only follows that R ∉ A. There's no contradiction and the theory is consistent.

In the second case, you consider classes, not just sets. Sets are classes that belong to some other class, while proper classes are classes that belong to no class. Set comprehension, in this case, says you can have {x : ϕ(x)}, but all its members are sets by definition. If try to reproduce Russell’s paradox, you get that R ∉ R. If you then suppose that R is a set, then you have a contradiction, so R must be a proper class. This is all you get. No contradictions. The theory is consistent.

SOLUTIONS TO THE PARADOX

Bertrand Russell devised what he called the theory of types to prevent the paradox. In this theory, a set would be defined as being of a distinct type, like type 1. The elements of type 1 sets can then only be included in a set of type 2 because sets of type 2 are defined as containing only sets of type 1. Thus, we do not need to worry about whether or not a set of type 2 can contain itself because it’s defined as only containing sets of type 1. This theory creates a sort of hierarchy of sets.

The most accepted solution today is that of Zermelo and Fraenkel. Zermelo’s axiom of specification is, "to every set A and every definite property P(x) there corresponds a set whose elements are exactly those elements x in A for which the property P(x) holds" (Burton, 616). What this axiom does is require a preexisting set A and some property P(x) to make a new set. Previously, only the property P(x) was required. This changes the set S to . Now is impossible because it would have to satisfy the two conditions that and , which it clearly cannot. The is clear because we just state that in order for , it must satisfy the property that . If we consider the other possibility that we see that it does satisfy the property P(x), but cannot meet our second requirement that . This is because if then it follows, by our definition, that , which we already reasoned is not true. Therefore we conclude that by the law of excluded middle, which says that every proposition is either true or false (Burton, 612), . Therefore, failed the second of the two requirements to be in the set S so we conclude that and have avoided the paradox.

Russell’s Early Responses to the Paradoxs

Russell’s own response to the paradox came with his aptly named theory of types. Believing that self-application lay at the heart of the paradox, Russell’s basic idea was that we can avoid commitment to R (the set of all sets that are not members of themselves) by arranging all sentences (or, more precisely, all propositional functions, functions which give propositions as their values) into a hierarchy. It is then possible to refer to all objects for which a given condition (or predicate) holds only if they are all at the same level or of the same “type.”

This solution to Russell’s paradox is motivated in large part by adoption of the so-called vicious circle principle. The principle in effect states that no propositional function can be defined prior to specifying the function’s scope of application. In other words, before a function can be defined, one must first specify exactly those objects to which the function will apply (the function’s domain). For example, before defining the predicate “is a prime number,” one first needs to define the collection of objects that might possibly satisfy this predicate, namely the set, N, of natural numbers.

As Whitehead and Russell explain,
          An analysis of the paradoxes to be avoided shows that they all result from a kind of vicious circle. The vicious circles in question arise from supposing that a collection of objects may contain members which can only be defined by means of the collection as a whole. Thus, for example, the collection of propositions will be supposed to contain a proposition stating that “all propositions are either true or false.” It would seem, however, that such a statement could not be legitimate unless “all propositions” referred to some already definite collection, which it cannot do if new propositions are created by statements about “all propositions.” We shall, therefore, have to say that statements about “all propositions” are meaningless. … The principle which enables us to avoid illegitimate totalities may be stated as follows: “Whatever involves all of a collection must not be one of the collection”; or, conversely: “If, provided a certain collection had a total, it would have members only definable in terms of that total, then the said collection has no total.” We shall call this the “vicious-circle principle,” because it enables us to avoid the vicious circles involved in the assumption of illegitimate totalities. (1910, 2nd edn 37)

If Whitehead and Russell are right, it follows that no function’s scope of application will ever be able to include any object presupposed by the function itself. As a result, propositional functions (along with their corresponding propositions) will end up being arranged in a hierarchy of the kind Russell proposes.

Although Russell first introduced his theory of types in his 1903 Principles of Mathematics, he recognized immediately that more work needed to be done since his initial account seemed to resolve some but not all of the paradoxes.

Possible Solutions to the Paradox of Classes or Sets

It was mentioned above that late in his life, Frege gave up entirely on the feasibility of the logic of classes or sets. This is of course one ready solution to the antinomy in the class or set form: simply deny the existence of such entities altogether. Short of this, however, the following solutions have enjoyed the greatest popularity:

The Theory of Types for Classes: It was mentioned earlier that Russell advocated a more comprehensive theory of types than Frege's distinction of levels, one that divided not only properties or concepts into various types, but classes as well. Russell divided classes into classes of individuals, classes of classes of individuals, and so on. Classes were not taken to be individuals, and classes of classes of individuals were not taken to be classes of individuals. A class is never of the right type to have itself as member. Therefore, there is no such thing as the class of all classes that are not members of themselves, because for any class, the question of whether it is in itself is a violation of type. Once again, here the challenge is to explain the metaphysics of classes or sets in order to explain the philosophical grounds of the type-division.

Stratification: In 1937, W. V. Quine suggested an alternative solution in some ways similar to type-theory. His suggestion was rather than actually divide entities into individuals, classes of individuals, etc., such that the proposition that some class is in itself is always ill-formed or nonsensical, we can instead put certain restrictions on what classes are supposed to exist. Classes are only supposed to exist if their defining conditions are so as to not involve what would, in type theory, be a violation of types. Thus, for Quine, while "x is not a member of x" is a meaningful assertion, we do not suppose there to exist a class of all entities x that satisfy this statement. In Quine's system, a class is only supposed to exist for some open formula A if and only if the formula A is stratified, that is, if there is some assignment of natural numbers to the variables in A such that for each occurrence of the class membership sign, the variable preceding the membership sign is given an assignment one lower than the variable following it. This blocks Russell's paradox, because the formula used to define the problematic class has the same variable both before and after the membership sign, obviously making it unstratified. However, it has yet to be determined whether or not the resulting system, which Quine called "New Foundations for Mathematical Logic" or NF for short, is consistent or inconsistent.

Aussonderung: A quite different approach is taken in Zermelo-Fraenkel (ZF) set theory. Here too, a restriction is placed on what sets are supposed to exist. Rather than taking the "top-down" approach of Russell and Frege, who originally believed that for any concept, property or condition, one can suppose there to exist a class of all those things in existence with that property or satisfying that condition, in ZF set theory, one begins from the "bottom up". One begins with individual entities, and the empty set, and puts such entities together to form sets. Thus, unlike the early systems of Russell and Frege, ZF is not committed to a universal set, a set including all entities or even all sets. ZF puts tight restrictions on what sets exist. Only those sets that are explicitly postulated to exist, or which can be put together from such sets by means of iterative processes, etc., can be concluded to exist. Then, rather than having a naive class abstraction principle that states that an entity is in a certain class if and only if it meets its defining condition, ZF has a principle of separation, selection, or as in the original German, "Aussonderung". Rather than supposing there to exist a set of all entities that meet some condition simpliciter, for each set already known to exist, Aussonderung tells us that there is a subset of that set of all those entities in the original set that satisfy the condition. The class abstraction principle then becomes: if set A exists, then for all entities x in A, x is in the subset of A that satisfies condition C if and only if x satisfies condition C. This approach solves Russell's paradox, because we cannot simply assume that there is a set of all sets that are not members of themselves. Given a set of sets, we can separate or divide it into those sets within it that are in themselves and those that are not, but since there is no universal set, we are not committed to the set of all such sets. Without the supposition of Russell's problematic class, the contradiction cannot be proven.

There have been subsequent expansions or modifications made on all these solutions, such as the ramified type-theory of Principia Mathematica, Quine's later expanded system of his Mathematical Logic, and the later developments in set-theory made by Bernays, Gödel and von Neumann. The question of what is the correct solution to Russell's paradox is still a matter of debate.

Russell's Theory of Types

Theory of types, in logic, a theory introduced by the British philosopher Bertrand Russell in his Principia Mathematica to deal with logical paradoxes arising from the unrestricted use of predicate functions as variables.
Arguments of three kinds can be incorporated as variables: (1) In the pure functional calculus of the first order, only individual variables exist. (2) In the second-order calculus, propositional variables are introduced. (3) Higher orders are achieved by allowing predicate functions as variables. The type of a predicate function is determined by the number and type of its arguments. By not allowing predicate functions with arguments of equal or higher type to be used together, contradictions within the system are avoided.

In the system of Russell and Whitehead, the formal objects of the theory are divided into types. The lowest type consists of all individuals, the next type is composed of all predicates, the succeeding type is composed of all predicates of predicates, and so on.

In order to avoid the paradox which he discovered, Russell formulated a Theory of Types. It was wrong to treat classes as randomly classifiable objects. Classes and individuals were of different logical types, and what can be true or false of one cannot be significantly asserted of the other. ‘The class of dogs is a dog’ should be regarded not as false but as meaningless. Similarly, what can meaningfully be said of classes cannot meaningfully be said of classes of classes, and so on through the hierarchy of logical types. If the difference of type between the different levels of the hierarchy is observed, then the paradox will not arise. (Antony Kenny”s A Brief History of Western Philosophy (1998 Blackwell Publishers Ltd), p325.)

Simple Type Theory

The types can be defined as
1.i is the type of individuals
2.( ) is the type of propositions
3.if A₁,…,A_n are types then (A₁,…,A_n) is the type of n-ary relations over objects of respective types A₁,…,A_n

For instance, the type of binary relations over individuals is (i, i), the type of binary connectives is (( ),( )), the type of quantifiers over individuals is ((i)).

For forming propositions we use this type structure: thus R(a₁,…,a_n) is a proposition if R is of type (A₁,…,A_n) and a_i is of type A_i for i = 1,…,n. This restriction makes it impossible to form a proposition of the form P(P): the type of P should be of the form (A), and P can only be applied to arguments of type A, and thus cannot be applied to itself since A is not the same as (A).

Russell observes, that it becomes necessary “to distinguish (1) terms, (2) classes, (3) classes of classes, and so on ad infinitum”. Moreover, these collections will be disjoint, and to able to assert x ∈ u requires that the collection to which x belongs should be of a degree one lower than that to which u belongs. This expedient leads to a resolution of the paradox, since x ∈ x has now been rendered a meaningless proposition. The hierarchy of collections (1), (2), (3), ... is the germ of the doctrine of types.

On Denoting

By a “denoting phrase” I mean a phrase such as any one of the following: a man, some man, any man, every man, all men, the present King of England, the present King of France, the centre of mass of the Solar System at the first instant of the twentieth century, the revolution of the earth round the sun, the revolution of the sun round the earth. Thus a phrase is denoting solely in virtue of its form. (Note: Denoting phrases include both definite descriptions and proper names.)
We may distinguish three cases: (1) A phrase may be denoting, and yet not denote anything; e.g., “the present King of France.” (2) A phrase may denote one definite object; e.g., “the present King of England” denotes a certain man. (3) A phrase may denote ambiguously; e.g., “a man” denotes not many men, but an ambiguous man.
The interpretation of such phrases is a matter of considerable difficulty; indeed, it is very hard to frame any theory not susceptible of formal refutation. All the difficulties with which I am acquainted are met, so far as I can discover, by the theory which I am about to explain.

The subject of denoting is of very great importance, not only in logic and mathematics, but also in theory of knowledge. For example, we know that the centre of mass of the Solar System at a definite instant is some definite point, and we can affirm a number of propositions about it; but we have no immediate acquaintance with this point, which is only known to us by description.
The distinction between acquaintance and knowledge about is the distinction between the things we have presentations of, and the things we only reach by means of denoting phrases.
It often happens that we know that a certain phrase denotes unambiguously, although we have no acquaintance with what it denotes; this occurs in the above case of the centre of mass. In perception we have acquaintance with objects of perception, and in thought we have acquaintance with objects of a more abstract logical character; but we do not necessarily have acquaintance with the objects denoted by phrases composed of words with whose meanings we are acquainted. To take a very important instance: There seems no reason to believe that we are ever acquainted with other people’s minds, seeing that these are not directly perceived; hence what we know about them is obtained through denoting. All thinking has to start from acquaintance; but it succeeds in thinking about many things with which we have no acquaintance.

My theory, briefly, is as follows.
I take the notion of the variable as fundamental; I use “C (x)” to mean a proposition in which x is a constituent, where x, the variable, is essentially and wholly undetermined.
Then we can consider the two notions “C (x) is always true” and “C (x) is sometimes true.” Then everything and nothing and something (which are the most primitive of denoting phrases) are to be interpreted as follows:
C (everything) means “C (x) is always true”;
C (nothing) means “ ‘C (x) is false’ is always true”;
C (something) means “It is false that ‘C (x) is false’ is always true.” 

Here the notion “C (x) is always true” is taken as ultimate and indefinable, and the others are defined by means of it. Everything, nothing, and something are not assumed to have any meaning in isolation, but a meaning is assigned to every proposition in which they occur. This is the principle of the theory of denoting I wish to advocate: that denoting phrases never have any meaning in themselves, but that every proposition in whose verbal expression they occur has a meaning.

Consider the proposition “all men are mortal.” This proposition is really hypothetical and states that if anything is a man, it is mortal. That is, it states that if x is a man, x is mortal, whatever x may be. Hence, substituting “x is human” for “x is a man,” we find:
        “All men are mortal” means “ ‘If x is human, x is mortal’ is always true.”

This is what is expressed in symbolic logic by saying that “all men are mortal” means “ ‘x is human’ implies ‘x is mortal’ for all values of x.” More generally, we say:
        “C (all men)” means “ ‘If x is human, then C (x) is true’ is always true.”

Similarly
“C (no men)” means “ ‘If x is human, then C (x) is false’ is always true.”
“C (some men)” will mean the same as “C (a man),” and
“C (a man)” means “It is false that ‘C (x) and x is human’ is always false.”
“C (every man)” will mean the same as “C (all men).”

It remains to interpret phrases containing “the”. These are by far the most interesting and difficult of denoting phrases.
Take as an instance “the father of Charles II was executed.”
This asserts that there was an x who was the father of Charles II and was executed.
Now “the”, when it is strictly used, involves uniqueness; we do, it is true, speak of “the son of So-and-so” even when So-and-so has several sons, but it would be more correct to say “a son of So-and-so.” Thus for our purposes we take “the” as involving uniqueness.
Thus when we say “x was the father of Charles II” we not only assert that x had a certain relation to Charles II, but also that nothing else had this relation.
The relation in question, without the assumption of uniqueness, and without any denoting phrases, is expressed by “x begat Charles II.”
To get an equivalent of “x was the father of Charles II,” we must add “If y is other than x, y did not beget Charles II,” or, what is equivalent, “If y begat Charles II, y is identical with x.”
Hence “x is the father of Charles II” becomes “x begat Charles II; and ‘if y begat Charles II, y is identical with x’ is always true of y.”

Thus “the father of Charles II was executed” becomes: “It is not always false of x that x begat Charles II and that x was executed and that ‘if y begat Charles II, y is identical with x’ is always true of y.”

This may seem a somewhat incredible interpretation; but I am not at present giving reasons, I am merely stating the theory.

To interpret “C (the father of Charles II),” where C stands for any statement about him, we have only to substitute C (x) for “x was executed” in the above. Observe that, according to the above interpretation, whatever statement C may be, “C (the father of Charles II)” implies:
        “It is not always false of x that ‘if y begat Charles II, y is identical with x’ is always true of y,” which is what is expressed in common language by “Charles II had one father and no more.” Consequently if this condition fails, every proposition of the form “C (the father of Charles II)” is false. Thus e.g. every proposition of the form “C (the present King of France)” is false. This is a great advantage to the present theory. I shall show later that it is not contrary to the law of contradiction, as might be at first supposed.

The above gives a reduction of all propositions in which denoting phrases occur to forms in which no such phrases occur. Why it is imperative to effect such a reduction, the subsequent discussion will endeavor to show.

The evidence for the above theory is derived from the difficulties which seem unavoidable if we regard denoting phrases as standing for genuine constituents of the propositions in whose verbal expressions they occur. Of the possible theories which admit such constituents the simplest is that of Meinong. This theory regards any grammatically correct denoting phrase as standing for an object. Thus “the present King of France,” “the round square,” etc., are supposed to be genuine objects. It is admitted that such objects do not subsist, but nevertheless they are supposed to be objects. This is in itself a difficult view; but the chief objection is that such objects, admittedly, are apt to infringe the law of contradiction. It is contended, for example, that the existent present King of France exists, and also does not exist; that the round square is round, and also not round, etc. But this is intolerable; and if any theory can be found to avoid this result, it is surely to be preferred.

The above breach of the law of contradiction is avoided by Frege’s theory. He distinguishes, in a denoting phrase, two elements, which we may call the meaning and the denotation. Thus “the centre of mass of the Solar System at the beginning of the twentieth century” is highly complex in meaning, but its denotation is a certain point, which is simple. The Solar System, the twentieth century, etc., are constituents of the meaning; but the denotation has no constituents at all. One advantage of this distinction is that it shows why it is often worth while to assert identity. If we say “Scott is the author of Waverley,” we assert an identity of denotation with a difference of meaning.

One of the first difficulties that confront us, when we adopt the view that denoting phrases express a meaning and denote a denotation, concerns the cases in which the denotation appears to be absent.
If we say “the King of England is bald,” that is, it would seem, not a statement about the complex meaning “the King of England,” but about the actual man denoted by the meaning.
But now consider “the King of France is bald.” By parity of form, this also ought to be about the denotation of the phrase “the King of France.” But this phrase, though it has a meaning provided “the King of England” has a meaning, has certainly no denotation, at least in any obvious sense. Hence one would suppose that “the King of France is bald” ought to be nonsense; but it is not nonsense, since it is plainly false.

Thus we must either provide a denotation in cases in which it is at first sight absent, or we must abandon the view that denotation is what is concerned in propositions which contain denoting phrases. The latter is the course that I advocate. The former course may be taken, as by Meinong, by admitting objects which do not subsist, and denying that they obey the law of contradiction; this, however, is to be avoided if possible. Another way of taking the same course (so far as our present alternative is concerned) is adopted by Frege, who provides by definition some purely conventional denotation for the cases in which otherwise there would be none. Thus “the King of France” is to denote the null-class. But this procedure, though it may not lead to actual logical error, is plainly artificial, and does not give an exact analysis of the matter. Thus if we allow that denoting phrases, in general, have the two sides of meaning and denotation, the cases where there seems to be no denotation cause difficulties both on the assumption that there really is a denotation and on the assumption that there really is none.

That the meaning is relevant when a denoting phrase occurs in a proposition is formally proved by the puzzle about the author of Waverley. The proposition “Scott was the author of Waverley” has a property not possessed by “Scott was Scott,” namely the property that George IV wished to know whether it was true. Thus the two are not identical propositions; hence the meaning of “the author of Waverley” must be relevant as well as the denotation, if we adhere to the point of view to which this distinction belongs. Yet, as we have just seen, so long as we adhere to this point of view, we are compelled to hold that only the denotation is relevant.

According to the view which I advocate, a denoting phrase is essentially part of a sentence, and does not, like most single words, have any significance on its own account. If I say “Scott was a man,” that is a statement of the form “x was a man,” and it has “Scott” for its subject. But if I say “the author of Waverley was a man,” that is not a statement of the form “x was a man,” and does not have “the author of Waverley” for its subject. Abbreviating the statement made at the beginning of this article, we may put, in place of “the author of Waverley was a man,” the following: “One and only one entity wrote Waverley, and that one was a man.” (This is not so strictly what is meant as what was said earlier; but it is easier to follow.) And speaking generally, suppose we wish to say that the author of Waverley had the property φ, what we wish to say is equivalent to “One and only one entity wrote Waverley, and that one had the property φ.”

The explanation of denotation is now as follows. Every proposition in which “the author of Waverley” occurs being explained as above, the proposition “Scott was the author of Waverley” (i.e. “Scott was identical with the author of Waverley”) becomes “One and only one entity wrote Waverley, and Scott was identical with that one”; or, reverting to the wholly explicit form: “It is not always false of x that x wrote Waverley, that it is always true of y that if y wrote Waverley y is identical with x, and that Scott is identical with x.” Thus if “C” is a denoting phrase, it may happen that there is one entity x (there cannot be more than one) for which the proposition “x is identical with C” is true, this proposition being interpreted as above. We may then say that the entity x is the denotation of the phrase “C.” Thus Scott is the denotation of “the author of Waverley.” The “C” in inverted commas will be merely the phrase, not anything that can be called the meaning. The phrase per se has no meaning, because in any proposition in which it occurs the proposition, fully expressed, does not contain the phrase, which has been broken up.

The puzzle about George IV’s curiosity is now seen to have a very simple solution. The proposition “Scott was the author of Waverley,” which was written out in its unabbreviated form in the preceding paragraph, does not contain any constituent “the author of Waverley” for which we could substitute “Scott.” This does not interfere with the truth of inferences resulting from making what is verbally the substitution of “Scott” for “the author of Waverley,” so long as “the author of Waverley” has what I call a primary occurrence in the proposition considered. The difference of primary and secondary occurrences of denoting phrases is as follows:

When we say: “George IV wished to know whether so-and-so,” or when we say “So-and-so is surprising” or “So-and-so is true,” etc., the “so-and-so” must be a proposition. Suppose now that “so-and-so” contains a denoting phrase. We may either eliminate this denoting phrase from the subordinate proposition “so-and-so,” or from the whole proposition in which “so-and-so” is a mere constituent. Different propositions result according to which we do. When we say “George IV wished to know whether Scott was the author of Waverley,” we normally mean “George IV wished to know whether one and only one man wrote Waverley and Scott was that man”; but we may also mean: “One and only one man wrote Waverley, and George IV wished to know whether Scott was that man.” In the latter, “the author of Waverley” has a primary occurrence; in the former, a secondary. The latter might be expressed by “George IV wished to know, concerning the man who in fact wrote Waverley, whether he was Scott.” This would be true, for example, if George IV had seen Scott at a distance, and had asked “Is that Scott?” A secondary occurrence of a denoting phrase may be defined as one in which the phrase occurs in a proposition p which is a mere constituent of the proposition we are considering, and the substitution for the denoting phrase is to be effected in p, and not in the whole proposition concerned. The ambiguity as between primary and secondary occurrences is hard to avoid in language; but it does no harm if we are on our guard against it. In symbolic logic it is of course easily avoided.

The distinction of primary and secondary occurrences also enables us to deal with the question whether the present King of France is bald or not bald, and generally with the logical status of denoting phrases that denote nothing. If “C” is a denoting phrase, say “the term having the property F,” then
“C has the property φ” means “one and only one term has the property F, and that one has the property φ.”

If now the property F belongs to no terms, or to several, it follows that “C has the property φ” is false for all values of φ.
Thus “the present King of France is not bald” is false if it means “There is an entity which is now King of France and is not bald,”
but is true if it means “It is false that there is an entity which is now King of France and is bald.”

That is, “the King of France is not bald” is false if the occurrence of “the King of France” is primary, and true if it is secondary. Thus all propositions in which “the King of France” has a primary occurrence are false; the denials of such propositions are true, but in them “the King of France” has a secondary occurrence. Thus we escape the conclusion that the King of France has a wig.

The whole realm of non-entities, such as “the round square,” “the even prime other than 2,” “Apollo,” “Hamlet,” etc., can now be satisfactorily dealt with.
All these are denoting phrases which do not denote anything.
A proposition about Apollo means what we get by substituting what the classical dictionary tells us is meant by Apollo, say “the sun-god.” All propositions in which Apollo occurs are to be interpreted by the above rules for denoting phrases. If “Apollo” has a primary occurrence, the proposition containing the occurrence is false; if the occurrence is secondary, the proposition may be true. So again “the round square is round” means “there is one and only one entity x which is round and square, and that entity is round,” which is a false proposition, not, as Meinong maintains, a true one.

“The most perfect Being has all perfections; existence is a perfection; therefore the most perfect Being exists” becomes:
“There is one and only one entity x which is most perfect; that one has all perfections; existence is a perfection; therefore that one exists.”
As a proof, this fails for want of a proof of the premiss “there is one and only one entity x which is most perfect.”

Mr. MacColl (Mind, N.S., No. 54, and again No. 55, page 401) regards individuals as of two sorts, real and unreal; hence he defines the null-class as the class consisting of all unreal individuals. This assumes that such phrases as “the present King of France,” which do not denote a real individual, do, nevertheless, denote an individual, but an unreal one. This is essentially Meinong’s theory, which we have seen reason to reject because it conflicts with the law of contradiction. With our theory of denoting, we are able to hold that there are no unreal individuals; so that the null-class is the class containing no members, not the class containing as members all unreal individuals.

It is important to observe the effect of our theory on the interpretation of definitions which proceed by means of denoting phrases. Most mathematical definitions are of this sort; for example “m − n means the number which, added to n, gives m.” Thus m − n is defined as meaning the same as a certain denoting phrase; but we agreed that denoting phrases have no meaning in isolation. Thus what the definition really ought to be is: “Any proposition containing m − n is to mean the proposition which results from substituting for ‘m − n’ ‘the number which, added to n, gives m.’ ” The resulting proposition is interpreted according to the rules already given for interpreting propositions whose verbal expression contains a denoting phrase. In the case where m and n are such that there is one and only one number x which, added to n, gives m, there is a number x which can be substituted for m − n in any proposition contain m − n without altering the truth or falsehood of the proposition. But in other cases, all propositions in which “m − n” has a primary occurrence are false.

The usefulness of identity is explained by the above theory. No one outside of a logic-book ever wishes to say “x is x,” and yet assertions of identity are often made in such forms as “Scott was the author of Waverley” or “thou are the man.” The meaning of such propositions cannot be stated without the notion of identity, although they are not simply statements that Scott is identical with another term, the author of Waverley, or that thou are identical with another term, the man. The shortest statement of “Scott is the author of Waverley” seems to be “Scott wrote Waverley; and it is always true of y that if y wrote Waverley, y is identical with Scott.” It is in this way that identity enters into “Scott is the author of Waverley”; and it is owing to such uses that identity is worth affirming.

One interesting result of the above theory of denoting is this: when there is anything with which we do not have immediate acquaintance, but only definition by denoting phrases, then the propositions in which this thing is introduced by means of a denoting phrase do not really contain this thing as a constituent, but contain instead the constituents expressed by the several words of the denoting phrase. Thus in every proposition that we can apprehend (i.e. not only in those whose truth or falsehood we can judge of, but in all that we can think about), all the constituents are really entities with which we have immediate acquaintance. Now such things as matter (in the sense in which matter occurs in physics) and the minds of other people are known to us only by denoting phrases, i.e. we are not acquainted with them, but we know them as what has such and such properties. Hence, although we can form propositional functions C (x) which must hold of such and such a material particle, or of So-and-so’s mind, yet we are not acquainted with the propositions which affirm these things that we know must be true, because we cannot apprehend the actual entities concerned. What we know is “So-and-so has a mind which has such and such properties” but we do not know “A has such and such properties,” where A is the mind in question. In such a case, we know the properties of a thing without having acquaintance with the thing itself, and without, consequently, knowing any single proposition of which the thing itself is a constituent.

Russell’s theory of descriptions

In his article On Denoting of 1905, Russell worked out a general theory of the meaning of definite descriptions, which would take care both of the cases where there was some object answering to the description (as in ‘the man who discovered oxygen’) and of the cases where the description was vacuous (as in ‘the present King of France’).

Russell saw that it was necessary to give an account of the meaning of vacous expressions such as ‘the present King of France’ when it occurred, for instance, in the sentence ‘the present King of France is bald’. Russell called expressions like ‘the man who discovered oxygen’ and ‘the present King of France’ by the name ‘definite description’.

Frege had treated definite descriptions simply as complex names, so that ‘The auther of Hamlet was a genius’ had the same logical structure as ‘Shakespear was a genius’. This meant that he had to provide for arbitrary rules to be laid down in order to ensure that a sentence containing an empty name or vacuous definite description did not lack a truth-value. Russell thought this was unsatisfactory, and proposed to analyse sentences containing definiye descriptions quite differently from those containing names. It is a mistake, he believed to look for the meaning of definite descriptions in themseves; only the propositions in whose verbal expression they occur have a meaning.

For Russell there is a big difference between a sentence such as “James II was deposed’ (containing the name ‘James II’) and a sentence such as ‘The brother of Charles II was deposed’. An expression such as ‘The brother of Charles II’ has no meaning in isolation; but the sentence ‘The brother of Charles II was deposed’ has a meaning none the less. It asserts three things:
a) that some individual was brother to Charles II
b) that only this individual was brother to Charles II
c) that this individual was deposed.

Or, more formally:
For some x, (a) x was brother to Charles II
and (b) for all y, if y was brother to Charles II, y=x
and (c) x was deposed.

The first element of this formulation says that at least one individual was a brother of Charles II, the second that at most one individual was a brother of Charles, so that between them they say that exactly one individual was brother to Charles II. The third element goes on to say that that unique individual was deposed.
In the analyzed sentence nothing appears which looks like a name of James II; instead, we have a combination of predicates and quantifiers.

What is the point of this complicated anslysis? To see this we have to consider a sentence which, unlike ‘The brother of Charles II was deposed’, is not true.

Consider the following two sentences”
(1) The sovereign of the United Kingdom is male.
(2) The sovereign of the United States is male.

Neither of these sentences is true, but the reason differs in the two cases.
Everyone would agree the first sentence is not true, but plain false, because, because the sovereign of the United Kingdom (i.e, Queen Elizabeth II) is female.
The second fails to be true because the Us has no sovereign, and on Russell’s view this second sentence is not just untrue but positively false; and consequently its ‘It is not the case that sovereign of th US is male’ is true.

(Editorial Note: Although both two sentence (1) and sentence (2) are false, sentence (1) obeys and sentence (2) violates the law of excluded middle, which states that for any proposition, either that proposition is true, or its negation is true. Here, the negation of (1) ‘The sovereign of the United Kingdoms is female’ becomes true, thus it obeys the law of excluded middle. On the other hand, the negation of (2) ‘The sovereign of the United States is female’ remains false, thus it violates the law of excluded middle.)

(Bertrand Russell's On Denoting: By the law of the excluded middle, either “A is B” or “A is not B” must be true. Hence either “the present King of France is bald” or “the present King of France is not bald” must be true. Yet if we enumerated the things that are bald, and then the things that are not bald, we should not find the present King of France in either list. Hegelians, who love a synthesis, will probably conclude that he wears a wig.)

According to Russell, sentences containing empty names, i.e. apparent names which name no objects, such as ‘Slawkenburgius was a genius,’ is not really a sentence at all, therefore neither true nor false, since there was never anyone of whom ‘Slawkenburgius’ was the proper name.

Why did ussell want to ensure that sentences containg vacuous definite descriptions should count as false? He was, like Frege, interested in constructing a precise and scientific language for purposes of logic and mathematics. Both Frege and Russell regarded it as essential that such a language should contain only expressions which had a definite sense, by which they meant that all sentences in which the expressions could occur should have a truth-value. For if we allow into our system sentences lacking truth-value, then inference and deduction become impossible. It is easy enough to recognize that ‘the round square’ denotes nothing, because it is obviously self-contradictory. But prior to investigation it may not be clear whether some complicated mathematical formula contains a hidden contradiction. And if it does so, we will not be able to discover this by logical investigation unless sentences containing it are assured of a truth-value.

The Theory of Description

According to the theory of descriptions, we must draw a distinction between proper names and what Russell called “definite descriptions”. A definite description is a phrase containing the word “the” in the singular, and it can be used to mention, refer to, or pick out exactly one person, thing, or place. A proper name seems to have the same function as a definite description; it always picks out or denotes a particular individual, and the individual it picks out is its meaning.
Thus, in the sentence, “Clinton is tall”, the term “Clinton” means the actual person, Clinton. Though definite descriptions and proper names may sometimes pick out the same individual or place, Russell argued that their logical functions are entirely disparate. Thus, a speaker who in 1996 asserted, “The President of the United State is tall”, might be using the definite description “the President of the United State” to refer to Bill Clinton, but that phrase is not Clinton’s name; it could be used on different occasions to refer to different individuals. If Clinton had been replaced as president in 1996 by another tall person, that phrase would refer to someone other than Clinton. Indeed, descriptive phrases can be used meaningfully without picking out anything. “The greatest natural number” does not – indeed cannot – pick out anything, since there is a strict proof that no such number can exist. “The present king of France”, if intended to refer to a twentieth-century monarch, also lacks a referent.

According to Russell, certain apparent names are not real names but abbreviated descriptions. “Hamlet”, “Medusa”, “Santa Claus”, and so on fall into this category; they are not the names of persons but appear in history via stories or literary accounts. In his play, Shakespeare gives us a description of a certain character; thus, in that play, the apparent name “Hamlet” is an abbreviation for a longer phrase such as “the main character in a play called Hamlet by William Shakespeare”. Once the distinction between proper names and descriptions is made, it can be demonstrated that sentences containing proper names and sentences containing descriptions (including apparent names) mean different things. This can be shown by translating the respective sentences into an ideal language, such as that of Principia mathematica, where the difference becomes perspicuous.

In the Principia, the rendering of sentences containing proper names and those containing definite descriptions takes a purely symbolic form. But the difference can also be expressed in English (which again shows how logic can capture the subtleties of ordinary discourse).
Thus, “Bill Clinton is tall”, is of the logical form “Fa”. This is a singular sentence, containing a logical constant “a” that stands for a proper name and a predicate term “F” that stands for a property. When the constant and the predicate are given descriptive meanings, as in the sentence “Clinton is tall”, both sentences ascribe a certain property to a particular individual. Both are thus logically singular sentences.
They can be contrasted with “The present king of France is tall”, which is grammatically a singular sentence but when translated into the notation of Principia is not of the form “Fa”. Rather, in English, it has the same meaning as “At least one person is a male monarch of France, at most one person is male monarch of France, and whoever is male monarch of France is tall”. It is thus not logically a singular sentence but a complex general one. In symbolic notation, it is expressed as a conjunction of three sentences, one of them asserting the existence of a French monarch:

1. (Ex)(MFx) (At least one thing is a male monarch of France).
2. (x)(y) ((MFx.MFy).(x=y)) (At most one thing is a male monarch of France).
3. (x) ((MFx.).Fx) (Whoever is a male monarch of France is tall).

In the English sentence, “The present king of France is tall”, the word “the” expresses singularity, referring to one object as monarch of France. Singularity (the concept of “the”) is captured by sentences (1) and (2). To say that one and only one object is king of France is to say that at least one such object exists and also that not more than one does. If there is such an object, then (1) and (2) are true; if the object has the property ascribed to it, then the whole sentence, “The present king of France is tall”, is true. If there is no such object, then (1) is false, and then “The present king of France is tall” is false. But if either true or false, it is necessarily meaningful. The combination of “at least one” and “not more than one” is equivalent to the notion of exactly one. This shows both how powerful and subtle an ideal logical language can be.

The use of a formal language to distinguish sentences containing names from those containing descriptions has a number of important implications for philosophy.
First, it shows that an ideal language not only can articulate the ordinary sentences of natural languages but also reveal distinctions that such languages conceal.
Second, this fact implies that one must distinguish surface grammar from a deeper logical grammar that expresses the real meaning of such sentences. According to this deeper grammar, definite descriptions are not names, and sentences containing definitive descriptions are not singular but general.
        This finding has direct philosophical import. For example, it clears up the puzzle of how an individual can consistently deny the existence of something. Suppose an atheist says, “God does not exist”. It would seem that the atheist is presupposing in these very words that there exists something, a God, that does not exist; the atheist seems to be contradicting him- or herself. Russell shows that in this sentence, “God” is not a name but an abbreviated description for (on a Judeo-Christian) “the x that is all powerful, all wise, and wholly benevolent”. The atheist’s sentence can now be read as saying: “There is nothing that is all powerful, all wise, and wholly benevolent”. The sentence thus allows a philosophical position to be expressed without falling into inconsistency. This result has similar implications for skepticism, as it allows a radical skeptic to deny that knowledge is attainable without presupposing that there is such a thing as knowledge.

Third, in the preceding analysis of the sentence, “The present king of France is tall”, the phrase “The present king of France” no longer appears as a single unit in any of the three sentences that taken together give its meaning. This means that the phrase “The present king of France” has been eliminated and replaced by a complex of quantifiers, variables, and predicates. If it were a proper name, it would not be eliminable. Because they are eliminable, definite descriptions are called :incomplete symbols” by Russell. His theory of descriptions is thus a theory about the nature and function of incomplete symbols. Finally, each of the analyzing sentence is a general sentence and each is meaningful. This fact is key to understanding how a sentence whose subject term lacks a referent can be meaningful.

In the light of the preceding account, we can summarize Russell’s objection to Meimong’s argument. Meinong essentially confused definite descriptions and names. Once “The present king of France” is seen to be a description, then the phrase need not refer to anything. Therefore, from the fact that a sentence containing the phrase is meaningful, it does not follow that its grammatical subject denotes anything. There is thus no need to posit the existence or subsistence of such entities as the present king of France, Hamlet, Medusa, or Santa Claus.

Russell’s theory of descriptions

In a simple subject-predicate statement such as “Socrates is wise,” Russell observed, there seems to be something referred to (Socrates) and something said about it (that he is wise). If the proper name in such a sentence is replaced by a “definite description”—as in the statement “The president of the United States is wise”—there is apparently still something referred to and something said about it. A problem arises, however, when nothing fits the description, as in the statement “The present king of France is bald.” Although there is apparently nothing for the statement to be about, one nevertheless understands what it says. Prior to Russell’s work on definite descriptions, some philosophers—most notably Alexius Meinong (1853–1920)—felt forced by such examples to conclude that, in addition to things that have real existence, there are things that have some other sort of existence, for such statements could not be understood unless there was something for them to be about.

In Russell’s view, philosophers like Meinong had been misled by the surface grammatical form of sentences containing definite descriptions. Although they treated them as if they were simple subject-predicate statements, in reality they were much more complex. Upon analysis, the statement “The present king of France is bald” is shown to be a complex conjunction of other statements. Rendered in symbolic logic, these statements are:
(i) (∃x)(Fx), or “There is a present king of France”;
(ii) (∀y)(Fy → y=x), or “There is at most one present king of France”; and
(iii) (∀x)(Fx → Bx), or “If anyone is a present king of France, he is bald.”
More important, each of the three component statements is general, in the sense that it does not refer to anything or anyone in particular. Thus, there is no phrase in the complete analysis equivalent to “the present king of France,” which shows that the phrase is not an expression, like a proper name, that refers to something as the thing that the whole statement is about. There is no need, therefore, to make Meinong’s distinction between things that have real existence and things that have some other kind of existence.

Because descriptions do not refer directly to things in the world, however, there must be some other way in which such a direct connection between language and the world is made. In search of this connection, Russell turned his attention to proper names. The name Aristotle, for example, does not seem to carry any descriptive content. But Russell argues, on the contrary, that ordinary names are really concealed definite descriptions (Aristotle may simply mean “The student of Plato who taught Alexander, wrote the Metaphysics, etc.”). If a name had no descriptive content, one could not sensibly ask about the existence of its bearer, for one could then not understand what is expressed by a statement involving it. If Russell were a name in this sense (without any descriptive content), then merely to understand the statement “Russell exists” or the statement “Russell does not exist” presupposes that one already knows what Russell refers to. But then there cannot be any genuine question about Russell’s existence, for just to understand the question one must know the thing to which the name refers. Ordinary proper names, however—Russell, Homer, Aristotle, and Santa Claus—as Russell pointed out, are such that it makes sense to question the existence of their bearers. Thus, ordinary names must be concealed descriptions and cannot be the means of directly referring to the particular things in the world.

Russell eventually concluded that things in the world can be talked about only through the medium of a special kind of name—in particular, one about which no question can arise whether it names something or not—and he suggested that in English the only possible candidates are the demonstrative pronouns this and that.

At this point in his thinking, Russell shifted from questions about the nature of language to questions about the nature of the world. He asked what sort of thing it is that can be named in the strict logical sense, that can be known and talked about, and from which one can learn about the world. The important restriction was that no question about whether it exists or not can arise. Ordinary physical objects and other people seemed not to fit this requirement.

In his search for something whose existence cannot be questioned, Russell hit upon present experience and, in particular, upon sense data: one can question whether one is really seeing some physical object—whether, for example, there is a desk before one—but one cannot question that one is having visual impressions or sense data. Thus, what a person can name in the strict logical sense and what things in the world he can refer to directly turn out to be elements of his present experience. Russell therefore made a distinction between what can be known by acquaintance and what can be known only by description—i.e., between things whose existence cannot be doubted and things about whose existence, at least theoretically, doubt can be raised. What is novel about Russell’s conclusion is that it was arrived at from a fairly technical analysis of language. To be directly acquainted with something is to be in a position to give it a name in the strict logical sense, and to know something only by description is to know only that there is something that the description uniquely fits.

Russell was not constant in his view about physical objects. At one point he thought that the observer must infer their existence as the best hypothesis to explain the observer’s experience. Later he held that they were “logical constructions” out of sense data.

Russell’s treatment of definite descriptions

Russell represented sentences of the form ‘The F is G’, e.g. ‘The present King of France is bald’, as conjoining three claims:
(1) There is at least one F (e.g. there is at least one present King of France).
(2) There is at most one F (e.g. there is at most one present King of France).
(3) Whatever is F is G (e.g. what is present King of France is bald).

In modern logical notation, the analysis becomes:
(4) (∃x) [Fx & (∀y) (Fy ⊃ x = y) & Gy].

By systematising a statement’s inferential (or, more broadly, logical) behaviour, the representation of the statement in a favoured logical system shows the (or, perhaps, a) logical form of the statement. Russell’s treatment provided a model on which a definite description that fails to apply to exactly one individual may be meaningful and so provided a potential solution to old problems about the functioning of talk that purports to make reference to particular non-existents. On Russell’s account, the fact that there is no present King of France makes the sentence ‘The King of France is bald’ false, rather than meaningless, due to its making false the first clause in his analysis. His treatment also made especially evident that the logical form of a statement might not be obvious from its superficial form. (However, the space between logical and superficial form involved in Russell’s treatment of ‘The F is G’ via (4) is an artefact of Russell’s favoured logic. An alternative, though slightly less perspicuous, treatment is given in (5):
(5) [The x: Fx] (Gx).)

Russell’s treatment of definite descriptions showed that philosophical progress could be made by discerning the (or a) logical form of a philosophically problematic range of statements and that some philosophical disputes are usefully viewed as (at least in part) concerning how best to represent the logical forms of statements involved in those disputes. Together with the new treatment of quantification more generally, became a model for a variety of approaches to philosophical problems that involved attention to the forms of language used in the statement of those problems. For it supported the view that philosophical problems can arise due to the misleading superficial forms of the language we use and provided a model for how problems that arise in that way might be solved through uncovering the true logical forms of the statements involved.

Russell’s Theory of Definite Descriptions

Russell’s philosophical method has at its core the making and testing of hypotheses through the weighing of evidence. Hence Russell’s comment that he wished to emphasize the “scientific method” in philosophy. His method also requires the rigorous analysis of problematic propositions using the machinery of first-order logic. It was Russell’s belief that by using the new logic of his day, philosophers would be able to exhibit the underlying “logical form” of natural-language statements. A statement’s logical form, in turn, would help resolve various problems of reference associated with the ambiguity and vagueness of natural language.

Since the introduction of the modern predicate calculus, it has been common to use three separate logical notations (“Px”, “x = y”, and “∃x”) to represent three separate senses of the natural-language word “is”: the is of predication, e.g. “Cicero is wise”; the is of identity, e.g. “Cicero is Tully”; and the is of existence, e.g. “Cicero is”. It was Russell’s suggestion that, just as we use logic to make clear these distinctions, we can also use logic to discover other ontologically significant distinctions, distinctions that should be reflected in the analysis we give of each sentence’s correct logical form.

On Russell’s view, the subject matter of philosophy is then distinguished from that of the sciences only by the generality and a prioricity of philosophical statements, not by the underlying methodology of the discipline. In philosophy, just as in mathematics, Russell believed that it was by applying logical machinery and insights that advances in analysis would be made.

Russell’s most famous example of his new “analytic method” concerns so-called denoting phrases, phrases that include both definite descriptions and proper names. Like Alexius Meinong, Russell had initially adopted the view that every denoting phrase (for example, “Scott,” “the author of Waverley,” “the number two,” “the golden mountain”) denoted, or referred to, an existing entity. On this view, even fictional and imaginary entities had to be real in order to serve as truth-makers for true sentences such as “Unicorns have exactly one horn.” By the time his landmark article, “On Denoting,” appeared in 1905, Russell had modified his extreme realism, substituting in its place the view that denoting phrases need not possess a theoretical unity. As Russell puts it, the assumption that every denoting phrase must refer to an existing entity was the type of assumption that exhibited “a failure of that feeling for reality which ought to be preserved even in the most abstract studies” (Introduction to Mathematical Philosophy, 165).

Russell’s most important writings relating to his theory of descriptions include not only “On Denoting” (1905), but also The Principles of Mathematics (1903), Principia Mathematica (1910) and Introduction to Mathematical Philosophy (1919). (See too “What is Russell’s Theory of Descriptions?” (Kaplan 1970), “Existence in the Theory of Definite Descriptions” (Kroon 2009), and “The Theory of Descriptions” (Stevens 2011). )

LOGIC AS THE IDEAL LANGUAGE

According to Russell, the system of the Principia was an extension of the ordinary language in the sense that it could capture its welter of different types of sentences and expose them to endless set of logical transformation, thus generating new theorems. It also represented a perfection of ordinary language by eliminating ambiguity and vagueness. But above all, because it was an instrument of razor sharpness, it could solve certain enduring philosophical problems.

Through its so-called theory of description, it could explain the invalidity of the ontological argument, which presupposed that existence was a property (or in Russell’s terms, a logical predicate). Thus, in the statements “Tigers growl” and “Tigers exist”, the words “grow;” and “exist” have different logical functions. The first means, “Something is a tiger and growls”, and the second means “Something is a tiger”. “Exists” is thus not a real predicate in the way that “growls” is. As the theory of descriptions demonstrates, however, existence functions as part of the apparatus of quantifications. Thus, the basic move in the ontological argument – that God is not perfect unless he possesses the property of existing – is fallacious because existing is not a property.

The theory of descriptions was able to resolve two other, deeper issues about existence and identity as well.

ABOUT EXISTENCE

From the time of the Greeks on, philosophers had puzzled about the nature of nonbeing without coming to any successful resolution of the issue. The problem can be stated thus: We are able to make significant and indeed sometimes even true statements about “entities” such as Santa Claus, Medusa, Hamlet, Atlantis, and so forth. It is surely true to say “Santa Claus does not exist”. Or, again, when we say, “Hamlet murdered Polonius”, that sentence seems to be true. But according to the standard correspondence theory of truth, a sentence p is true if and only if it corresponds to a particular fact in the world. Thus, the world does not contain the fact that Hamlet murdered Polonius, since in reality that putative event never occurred. Moreover, on the most simple and intuitive theory of language, it seems plausible to hold that words obtain their meanings because they correspond to certain sorts of objects. Thus, the word “dog” in the sentence “Some dogs are white” is meaningful because there are objects in the world – namely, dogs – that it picks out or denotes. Yet “Santa Claus”, “Hamlet”, and “Atlantis” all seem to be meaningful, even though they denote no existing things.

In the twentieth century, the problem of nonbeing surfaced in the work of the Austrian logician Alexius Meinong (1853-1920), who advanced the thesis that “there are objects that do not exist”. In 1904, Russell accepted this theory, but by 1907 he had rejected it. Meinong argued that such things as the Fountain of Youth, the present king of France, Santa Claus, and Hamlet – which ordinary people regard as nonexistent – must exist in some sense or another. The special sense he called Bestand (subsistence).
Meinong was led into this position by an argument that can be rephrased as follows: (1) The phrase “the present king of France” is the subject of the sentence, “The present king of France is wise”. (2) Since the sentence “The present king of France is wise” is meaningful, it must about something – namely, the present king of France. (3) But unless the king of France existed, the sentence would not be about anything and hence would not be meaningful at all, since one of its essential constituents, “The present king of France”, would not be meaningful. (4) Sine “The present king of France is wise” is meaningful, it therefore must be about some entity – namely, the present king of France – hence, such entity must exist or subsist.

For Russell, this argument not only was fallacious but it lacked – as he put it – the “robust sense of reality” that one should expect in good philosophy. Santa Claus is not a creature of fresh and blood, and no object is now or was ever king of France in the twentieth century. The fallacy in the argument was exposed via the theory of descriptions.

ABOUT IDENTITY

The new logic was also able to solve long-standing problems about the nature of sameness or identity. This issue is central to a number of major problems, among them the ancient problem of change that puzzled the Greeks and the problem of personhood that bothered seventeenth- and eighteenth-century thinkers. Frege and Russell independently invented different ingenious solutions to the problem. There is a serious debate within the philosophy of language over which solution is preferable and each is widely accepted today. Among the important contemporary writers who have contributed to the debate are Quine, Searle, Ruth Marcus, Keith Donnellan, Saul Kripke, Hillary Putnam, and David Kaplan.

Frege presented his solution in a paper, “über Sinn und Bedeutung” (On Sense And Reference) that was originally published in 1893 and received little recognition in its own time but was rediscovered after World War II and has been influential ever since. Frege begins by stating that the idea of sameness challenges reflection. He formulates the problem thus: Consider two true identity sentences, “Venus = Venus” and “Venus = the morning star”. The first is trivial, a tautology that communicates no new information. The second, however, seems to represent an extension of our knowledge. But if both sentences refer to the same object – namely, a specific planet – how can the second sentence be significant while the first is not? Are we not referring to the same object twice over and thus merely repeating ourselves?

Frege solved this problem by drawing a distinction between two senses of “meaning”. Linguistic expressions, he stated, have meaning in a referential or extensional sense (bedeutung) in which they refer to a particular object, in this case the planet Venus. But they also have a connotative or intensional meaning (sinn), in which they may allude to the object indirectly, via a description of it. With this distinction in hand, the two identity sentences clearly differ in significance. In stating that Venus is the morning star, we add new information; namely that this is the planet that appears in the morning sky. Everyone knows a priori that Venus is Venus, but it was an important astronomical discovery that Venus is the planet that appears in the morning sky. The knowledge that one is referring to the same planet under a special description makes the sentence significant and not trivial. Frege’s solution was that the term “Venus” and “the morning star” are identical in meaning in the extensional but not in the intensional sense, and it is the latter difference that makes the second sentence significant. Frege generalized this brilliant insight into an entire philosophy of language that applied not only to words but to larger units of language as well, such as descriptions and sentences.

Russell, however, denied that genuine proper names such as “Venus” possess intensional meaning. According to him, they mean only the object they denote. His solution to the problem is that because the phrase “the morning star” is a definite description, the sentence “Venus = the morning star” is not an identity sentence at all but a complex general sentence that should be analyzed according to the theory of definite descriptions.

Frege’s account is generally supported on the ground that it captures the grammatical form of the English sentences, allowing both to be identity sentences, but it has the disadvantage – as Quine, Kripke, and Putnam have emphasized – that intensions or senses are not well-defined entities.
Russell’s account treats names as kinds or tags that directly pick out an object without the mediation of a description or intension. This treatment of names has received widespread acceptance, but Russell’s account has the disadvantage that it analyzes what seems prima facie to bean identity sentence into a set of sentences of a completely different logical form.
These differing approaches have generated a vast contemporary literature in which merits and disadvantages of each theory have been extensively probed.