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I am not quite sure I interpret the following sentence correctly in Bertrand Russell's paper on existential import:

and among classes there is just one which does not exist, namely, the class having no members, which is called the null-class.

This seems to be saying that the empty set does not exist, and although I would agree with that, it seems to me that it very unlikely to be what he meant.

So my question is:

What did Russell mean when he wrote that the null-class, the class he defined as having no members, did not exist?

I put here the relevant passage which is on his distinction between what he sees as the two meanings of the word "existence":

The first point in regard to which clearness is essential concerns the meaning of the word “existence.” There are two meanings of this word, as distinct as stocks in a flower-garden and stocks on the Stock Exchange, which yet are continually being confused, or at least supposed somehow connected. Of these meanings, only one occurs in philosophy or in common parlance, and only the other occurs in mathematics or in symbolic logic. Until it is realised that they have absolutely nothing to do with each other, it is quite impossible to have clear ideas on our present topic.

(a) The meaning of existence which occurs in philosophy and in daily life is the meaning which can be predicated of an individual, the meaning in which we inquire whether God exists, in which we affirm that Socrates existed, and deny that Hamlet existed. The entities dealt with in mathematics do not exist in this sense: the number 2, or the principle of the syllogism, or multiplication, are objects which mathematics considers, but which certainly form no part of the world of existent things. This sense of existence lies wholly outside Symbolic Logic, which does not care a pin whether its entities exist in this sense or not.

(b) The sense in which existence is used in symbolic logic is a definable and purely technical sense, namely this: To say that A exists means that A is a class which has at least one member. Thus whatever is not a class (e.g., Socrates) does not exist in this sense; and among classes there is just one which does not exist, namely, the class having no members, which is called the null-class. In this sense, the class of numbers (e.g.) exists, because 1, 2, 3, etc., are members of it; but in sense (a) the class and its members alike do not exist: they do not stand out in a part of space and time, nor do they have that kind of super-sensible existence which is attributed to the Deity.

Bertrand Russell, The Existential Import of Propositions (1905)

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    Classes were extensional counterparts of Aristotle's predicates during the transition from Boole to Zermelo, see Kanamori. Syllogism premises are often phrased as "some B-s are A-s", and "A-s exist" was awkwardly translated into "class A exists" in Russell's b-sense. As already Boole wrote, "the meaning of the term will be extended so as to include... the cases denoted by the terms 'nothing' and 'universe'", and so there is a (single) special class N with no N-s existing. In 1908 Zermelo switched to N "existing" despite having "no elements at all".
    – Conifold
    Commented Feb 9, 2023 at 11:46
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    Russell in 1905 was using a definition of existence in mathematics which is no longer current. Most modern mathematicians regard the empty set as existing in the same way that other sets exist. Indeed, in the von Neumann definition of ordinals, the number zero is the empty set. Russell's definition of numbers in terms of equivalence classes runs into the problem that these equivalence classes are too big to be sets, i.e. they are proper classes....
    – Bumble
    Commented Feb 9, 2023 at 16:16
  • See, e.g. this question: math.stackexchange.com/questions/3178282/…. I am aware of a few mathematicians who do not accept that the empty set exists, and do not accept that the number zero exists. The two usually go together.
    – Bumble
    Commented Feb 9, 2023 at 16:16

7 Answers 7

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Let me start by slightly rephrasing what Russell wrote, since Russell is using the word "exists" in an unusual and confusing way.

With my changes in bold, here is what Russell wrote:

(b) The sense in which inhabitancy is used in symbolic logic is a definable and purely technical sense, namely this: To say that A is inhabited means that A is a class which has at least one member. Thus whatever is not a class (e.g., Socrates) is not inhabited in this sense; and among classes there is just one which is not inhabited, namely, the class having no members, which is called the null-class. In this sense, the class of numbers (e.g.) is inhabited, because 1, 2, 3, etc., are members of it; but in sense (a) the class and its members alike do not exist: they do not stand out in a part of space and time, nor do they have that kind of super-sensible existence which is attributed to the Deity.

For the next few minutes, please join me in pretending that Russell wrote the above paragraph exactly as it's shown here.

Russell explains exactly what he means by the phrase "is inhabited": namely, he means "is a class which has at least one member." Russell writes that the null-class is not inhabited, and what he means when he writes this is that the null-class is not a class that has at least one member. And, of course, that statement is completely accurate: the null-class is a class, but it doesn't have any members.

Moreover, Russell writes that the null-class is the only class that is not inhabited. And what Russell means by this is that every class (except for the null-class) has at least one member.

I hope that all of this is more or less straightforward. I don't think there's anything strange or counterintuitive about the claim that the null-class (in other words, the empty set) is uninhabited.

Okay, we can stop pretending now, and let's discuss what Russell actually wrote.

Everything that I wrote above is true, except for one detail: namely, Russell actually wrote "exists" instead of "is inhabited." But that detail is completely insignificant. It doesn't matter at all what word Russell used when describing this concept. He could have used the phrase "has pink toenails" and the paragraph would mean exactly the same thing.

The problem with Russell's choice of words is that the word "exists" already means something else, and so when we read the word "exists" as Russell uses it in paragraph (b), we tend to assume that the word "exists" as Russell uses it is in some way similar to the word "exists" that we are already familiar with. But that assumption is false; Russell's "exists" is a completely, totally, and utterly different concept from "exists" in the usual sense. (Granted, it is a related concept, but it's not a similar concept.)

You ask if it would be accurate to say that according to Russell, the null-class "does not exist at all." I definitely would not say that, because the phrasing "does not exist at all" mixes the two dissimilar senses of the word "exists" with each other in a way that doesn't make sense. Saying "the empty set does not exist at all" to mean "the empty set doesn't exist in the everyday sense, and it doesn't exist in the sense of Russell's paragraph (b) either" would be like saying "my dog doesn't have any arms at all" to mean "my dog doesn't have any limbs with hands on the end of them, and he doesn't have any weapons either."

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  • +1 This is the only answer that actually addresses Russel's issue in a clear way
    – Neil_UK
    Commented Feb 10, 2023 at 8:30
  • @TannerSwett Thanks, excellent answer, both in substance and in form. Congratulation, you win the tick. And Russell deprived himself of the possibility of saying that the empty set existed. Funny. Commented Feb 10, 2023 at 17:27
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    Beautifully put. I would add one nuance to all of this: Russell demonstrates that his collection is inhabited by constructing one of its elements. This is the difference between "inhabited" and "non-empty" for constructivists, and may be helpful when reading other constructivists doing set theory.
    – Corbin
    Commented Feb 10, 2023 at 21:37
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    +1. Just to add a little: in 1905 when Russell wrote this, the terminology of formal logic and especially set theory was still very much in flux. People were just beginning to nail down what the central formal concepts of set theory were, and following that, what ordinary terms to use for them. It a writer after about 1925 defined “existence” in a way clashing with the modern standard sense, we’d call it idiosyncratic at best — but from Russell in 1905, it’s not unreasonable at all. Commented Feb 12, 2023 at 16:06
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To say that A exists means that A is a class which has at least one member. Thus whatever is not a class (e.g., Socrates) does not exist in this sense.

To say the Null Class exists means that the Null Class is a class which has at least one member. This is a contradiction.

It's a consequence of how Russell has defined "exists"

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In the passage above Russell discusses two uses of existence:

(a) is the "common sense" use: "which occurs in philosophy and in daily life is the meaning which can be predicated of an individual" [space-time existence], and

(b) "the sense in which existence is used in symbolic logic [in] a purely technical sense, namely this: To say that A exists means that A is a class which has at least one member.

This second sense applies to concept: we say that the concept "mountain" is instantiated, while we say that the concept "unicorn" is not.

Thus, due to the fact that there are no unicorns in our world, the concept "unicorn" is not instantiated and thus the class of unicorns is empty.

Using the second sense of "existence", Russell would say that the class Unicorn does not exist.

See also the reference to abstract objects, i.e. numbers:

In this sense [(b)], the class of numbers exists, because 1, 2, 3, etc., are members of it; but in sense (a) the class and its members alike do not exist: they do not stand out in a part of space and time.

Thus classes do not exist in sense (a), being abstracts. The null-class is an abstract and, being empty, does not exist neither in sense (a) nor in sense (b):

in sense (b), which is alone relevant [for logic: "the only sense in which symbolic logic is concerned with realities"], there is among classes not a multitude of non-existences, but just one, namely, the null-class.


The context of the discussion with Hugh MacColl is relevant also: the Existential Import of propositions.

See also MacColl's reply to Russell:

As regards Existential Import, the one important point on whioh I appear to differ from all other symbolists is the following. The null class o, which they define as containing no members, and which I, for convenience of symbolic operations, define as consisting of the null or unreal members o1,o2, etc., is understood by them to be contained in every class, real or unreal; whereas I consider it to be excluded from every real class.

Here we can see the key-point of the debate: Russell is defending against MacCall the (now) current point of view: the the null-class (empty set) is included in every set.

Their convention of universal inclusion leads to awkward and, I think, needless paradoxes, as, for example, that "Every round square is a triangle," because round squares form a null class, which, by them, is understood to be contained in every class. My convention leads, in this case, to the directly opposite conclusion, namely, that "No round square is a triangle," because I hold that every purely unreal class, such as the class of round squares, is necessarily excluded from every purely real class, such as the class of figures called triangles.


See also Principles of Mathematics (1903), §25:

Another very important notion is what is called the existence of a class— a word which must not be supposed to mean what existence means in philosophy. A class is said to exist when it has at least one term.[...] A new feature of the class-calculus is the null-class, or class having no terms. This may be defined as the class of terms that belong to every class, as the class which does not exist (in the sense defined above), as the class which is contained in every class.

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    Are you saying that what Russell means is that the null-class doesn't exist in sense (a) but does in sense (b)? This does not seem to fit with what he says: among classes there is just one which does not exist, namely, the class having no members, which is called the null-class. Thus, the null-class is the only one which doesn't exist. This is not true in sense (a) because of unicorns and such, so he must be talking about sense (b), which would be consistent with the fact that he is there discussing sense (b). Commented Feb 9, 2023 at 8:34
  • We may say that sense (a) is that used in "President Biden exists" that can be formulated as ∃x(PresB=x), while sense (b) must be applied to class and we may formulate is as: Ex[(^xP(x)] iff ∃xP(x) where the LHS uses a different operator for the existence of the class corresponding to predicate P and the RHS holds when predicate P is instantiated. – Commented Feb 9, 2023 at 10:23
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    I think that @Speakpigeon 's confusion comes from this part of your answer, which is probably not what you intended to write: "[…] the null-class […] exists in sense (b) but not in sense (a)"
    – Oliphaunt
    Commented Feb 9, 2023 at 19:18
  • @Oliphaunt - thanks you very much. Commented Feb 9, 2023 at 20:40
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The definition that Russell provides for (b):

To say that A exists means that A is a class which has at least one member.

We translate this into modern mathematical notation as follows:

"A exists" ↔ ∃x x ∈ A

This sense of the word "exists" is not the same as the modern ∃ symbol, because it is applied to A and not x. In modern mathematical parlance, we would instead say that the members of A exist.

If you look at that formula, you will see that the empty set fails to satisfy it (because there is no x that is a member of the empty set - in other words, it has no members that can be said to exist). However, the empty set's singleton would satisfy this definition, because we can have x = the empty set.

Russell is not claiming that the empty set fails to "exist" in the modern mathematical sense of the word. He is instead using the word with a slightly different definition, that happens to exclude the empty set.

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    "modern mathematical parlance" You use this phrase twice, and obviously to mean two different things, maybe you could decide which is really modern mathematical parlance? 2. "This sense of the word "exists" is not the same as the modern ∃ symbol" Sure but the question is about Russell's sense. 3. Your answer does not address my question. Commented Feb 9, 2023 at 14:14
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    @Speakpigeon: You asked whether Russell is claiming that the empty set fails to exist. This is answered by the final paragraph; the rest is providing context.
    – Kevin
    Commented Feb 9, 2023 at 20:22
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I daresay Russell is simply contradicting himself, per the broader context of his analysis of set theory, because the passage you quote comes from a text published in 1905, but in 1901 he had already come upon the paradox of the Russell set, which by his lights doesn't exist either, and yet which "if" it existed surely would not be the null class.

And the "way around" the Russell paradox, that he favored, i.e. his theory of types that are constrained by predicativist considerations, wipes out an unrestricted set simpliciter, not just a set with an unrestricted intensional parameter for noncircular sets. So Russell would have to have it that even when it comes to the abstract usage of the word "existence," there are at least two sets besides the empty one, that do not exist; and more pointedly, his position might have required that any comprehension axiom which generates what we nowadays usually call "proper classes" instead generates a nonexistent set (or, better, such an axiom simply does not generate a set, and the counterpossible set that it "could have" generated then fails to exist).

Alternatively, though, perhaps Russell would have just thought to conflate all nonexistent sets with the empty one in the limit. He might have thought to "parody" Frege's introduction of an empty set, which is the claim that there is a set that contains all violations of the law of identity, i.e. which contains nothing (because nothing violates the law of identity). This was an intensionalist move; for extensionalists, it is enough to just zero out a set's elements to make it into an empty one, there's no need for a convoluted "proof" that some set is empty owing to a weird comprehension axiom. At any rate, then, perhaps Russell was minded to think that the empty set is empty because it is a set that violates the law of noncontradiction, and so not only is it empty, but the rest of reality is empty of it, in that it does not exist even in terms of abstract existence.

Many times in his life, Russell involved himself in valiant causes: opposing the holocaust in the Congo Free State, opposing the holocaust in Vietnam, opposing the possibility of nuclear holocaust across the face of the Earth. Intellectually, he aimed for epistemic virtue; but he was also something of a polemicist, and I think the sentiments/attitudes underlying his polemicism might have led him to make "provocative" assertions in various contexts. But for that, these assertions might not deserve to be taken very seriously. If he has already denied one form of existence, let's call it X-existence, to abstract number-objects, but saw fit to assign a different form of existence, Y-existence, to those things nevertheless, then he might as well have gone on to think up a third form of existence, Z-existence, which things that are X- or Y-nonexistent happen to have. In principle, modulo his type theory, he could have gone on to assert that there are infinitely many forms of existence, one, at least, for every reach of his ramified type theory. I don't know the details of his type theory all too well, and what I do know indicates that he didn't really have a perfectly settled such theory (perfectly so even in his own eyes, that is), but so still, I would suggest that it is possible that Russell "drastically" misspoke in the passage you quoted.

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Russel is illustrating the meaning of a particular, specialised use of the word 'exist' in mathematics. By definition, the null class does not 'exist' in that particular meaning of the word, which Russell defined as applying only to sets that have at least one member.

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    "By definition, the null class does not 'exist' in that particular meaning of the word" Sure but it does not exist either in the sense (a) used in philosophy and everyday language, so it does not exist at all. Is this really what you think he meant? Commented Feb 9, 2023 at 8:24
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    I think you are painting yourself into a corner. Russell is saying that the null class does not exist in the way that trees, cars, London etc exist, and he is saying that it does not exist in the sense that set theorists mean. However, the idea of an empty set, such as the set of round squares, 'exists' as an idea. So, as with most problems in philosophy, the questions boils down to a lack of precision in the use and meaning of words. Commented Feb 9, 2023 at 9:46
  • Again questions about "existence".
    – Frank
    Commented Feb 9, 2023 at 17:28
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    @MarcoOcram But set theorists do say that the null class exists, in the same sense that any other class exists. Russell's terminology is outdated and does not reflect current mathematical practice. It is important to clarify this.
    – user76284
    Commented Feb 9, 2023 at 19:56
  • @user76284 thanks, that's a useful clarification Commented Feb 9, 2023 at 19:57
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Think of it this way ... just ss 0 is an abstraction of the number of elves, 10-headed dogs, dodos, etc., the null set is an abstraction of null classes. If 0 exists as a number, so too does the null set as a set.

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