For instance,
There does not exist unicorn on earth. There are zero unicorns on earth.
How are these two different?

One could argue being zero in quantity is a temporal property. The number may change with time.
But, is concept of time there in the picture when we are making the statement. The statement doesn't mention anything about it. I just know about an object unicorn and I'm only concerned about how many such are there right now.

I'm not sure what tags would suite the question, so please remove or add accordingly.

  • A body at rest has zero speed; does this mean that its speed is "nothing"? Nothing is a tricky term. Oct 20, 2022 at 15:43
  • I think 'to exist' is a difficult thing to specify, as said below, 'unicorn' as an idea does indeed exist, it's just there does not exist any instances that fall under it, you could argue the fact we have an 'idea' of unicorn implies the existence of a set, one that is simply empty. Some may argue that such a set/class does not exist,I would say that they are equivalent but we must be careful how we define 'existence'. Stating that there are no elements that can fall under such a set is equally true in both cases, and in your use of 'existence' makes sense.
    – Confused
    Oct 20, 2022 at 17:09
  • Existence can change in time just like the number, there were no lasers and microchips in the 19th century, but there sure are now. And why should no A's be different from 0 A's except in emphasis and context?
    – Conifold
    Oct 21, 2022 at 1:48
  • 1
    There is a difference between "impossible to exist" and "not existing but can exist"
    – Nikos M.
    Oct 21, 2022 at 6:39

5 Answers 5


A number applies to a class, rather than to an individual. The number three doesn't apply to each individual vertex of a triangle, rather the number three applies to the class of all of the vertices of a given triangle. Even if there is only one object, the number applies to the class of those objects, not to the individual. When you say there is exactly one current president of the US, the number one does not apply to the individual named "Joe Biden"; it applies to the class expressed by "current president of the US".

Many (probably most) philosophers would claim that by contrast, existence is something that applies to an individual. For example, George Washington no longer exists. This is a statement about the individual George Washington. However, there is a problem with this account, and that is that it is hard to assign a semantics to a sentence like that. If George Washington doesn't exist, then what does the name "George Washington" refer to? If it refers to nothing, then the sentence reduces to "nothing doesn't exist", which is a bit odd.

To deal with this and other problems of the view that existence is a feature of individuals, some philosophers claim that existence is just like number: it applies to classes rather than to individuals. For example, when you say that are no unicorns, you are saying that the class of unicorns is empty.

This doctrine has to account for existence statements of the form "Joe Biden exists" which seems to be about an individual named "Joe Biden". One way to deal with this is to say that the sentence is really about the concept of Joe Biden rather than the individual Joe Biden. The concept of Joe Biden has a class of individuals that instantiate that concept. The class has one member. Thus, to say that Joe Biden exists is to say that there is something that instantiates the concept of being Joe Biden. This works equally well with George Washington who does not (currently) exist. To say that George Washington does not exist is to say that the concept of being George Washington is not instantiated.

Of course, there are many arguments on both sides of the issue, which I won't go in to here. However, with that background, here is the short answer to your question:

If you believe that existence is a property of individuals, then number and existence are not the same thing because number applies to a class and existence applies to an individual. Number is about a class of which the individual is a member, existence is something that the individual has. If you believe that existence is about classes rather than individuals, then, yes, saying that there are zero unicorns is pretty much the same as saying that unicorns do not exist, in the same way that saying a set has zero members is the same as saying the set is empty.

  • The concept of forming a set/class of objects does exist intrinsically as soon as the object is thought of, in the same way that the concept (and definition) of unicorn exists in our thought. But that existence in thought is different from physical existence, right? So what do you mean by "If you believe that existence is a property of classes" ? Oct 21, 2022 at 6:35
  • @ParthBhagwat, I meant the second approach to existence that I explained in the first part of the answer. I've changed the wording in an attempt to clarify. Oct 21, 2022 at 8:48
  • I'd read Russel's argument as when we say Unicorn's don't exist, then what is it that is not existing? There should be something with which we are associating the word Unicorn. Of course we have defined it as horse with a horn on forehead and there can be found many toys/images of the creature. But, if I define Unicrow(a crow with a horn on forehead) and make a similar statement of non-existence. This creature is just in my thought (and the reader's), but has not been physically represented. Will the statement hold now? Since existence in thought is different from physical existence, right? Oct 24, 2022 at 8:21
  • 1
    @ParthBhagwat, I would say that there is a concept of a unicorn which is different from the unicorn. It is the concept which exists in thought--and the concept does exist. However, if the unicorn exists, it doesn't exist in thought; it exists in a forest. Oct 24, 2022 at 14:57

The difference is the container:

  • A non-existent object is a lack of existence, a full negative without positive associated entities.

  • Zero objects implies the existence of a countable set, which is empty.

In the first case, there is no set.

The comparison is equivalent to this two forms of the same concept:

a) the non-existence of a bunch of sand.

b) a hole (evidently, nothing inside, not even sand).


There does not exist unicorn on earth. There are zero unicorns on earth. How are these two different?

They are not different, they are equivalent. Certainly in this context since unicorns are imaginarinary but even statements like:

There does not exist any water on earth. There is zero water on earth.

are still equivalent. There is never any condition where one is true and the other is false.

The statements: "Water never existed on earth" and "There is zero water on earth" are different since some future date water may be gone (no water on earth) but water did exist at some time.


Consider the difference between two types of zero-theoretic graphs:

  1. A graph with no nodes at all.
  2. A graph with at least one node, but this node has no arrows, either from itself to itself, or from itself anywhere else; neither from anywhere else to the initial node.

Note that it is possible to contrive a parafounded hierarchy of sets starting out with a Quine atom, a set which has itself as its sole element. So in parafounded set theory, we can have a Quine origin or, if you work with (2) per the above, a normal "empty set" as the origin instead. Yet apparently there is a deeper representation of "nothingness," here, then, viz. a graph of type (1). (C.f. C. S. Pierce's use of blank pages as symbols of truth, not falsity/emptiness, however.)

You can also differentiate between notions like empty sequents and empty vocabularies (they mention these about a dozen times in the linked article, although the basic definition eludes me). Consider also the concept of ur-elements but also, then, the concept of that which is neither an element nor a set at all (a sort of "absolute" zero). You might represent it as the difference between:

  1. For all n (including 0), A does not equal n.
  2. A = at least 0.

Inasmuch as the concept of existence is used as a quantifier, it is interchangeable enough with a quantifier juxtaposed with the concept of zero or "none," i.e., "X does not exist," goes to, "There exist 0-many X." However, it is unclear whether existential quantification is all there is to existence, or how the term actuality in modal logic figures in the overall picture (for example). So consider the difference between something being actual and therefore "obviously" possible, and something being merely possible; yet are being merely possible and being nonexistent identical? But perhaps we would have to always qualify "existent/nonexistent" modally, then. So then what is the difference between, "There are actually 0-many of such-and-such a thing," and, "There are possibly 0-many of such-and-such a thing," or even worse, "There are 0-many possible such things," and, "There are 0-many actual such things"?

  • Oh man, the site had me check to make sure I'm not a bot. Like, did I not give off bot vibes in other posts? Why this one? Oct 21, 2022 at 16:43
  • Considering a class/set of the objects, as David Gudeman's answer suggests, seems correct when saying something as "quantity of unicorns is zero" equivalent to "there does not exist an unicorn". But I'd encountered an argument as whether non existence be quantified in any manner? Is zero a quantification of non existence?(PS: I haven't read the linked articles/pages yet.) Oct 22, 2022 at 4:27

Generally, yes- saying there are no unicorns is equivalent to saying unicorns do not exist.

However, in everyday language we are more likely to say something does not exist when we were speaking in a more general sense, and say there is none of something when we are speaking in a more specific sense. For example, I might say that I went to the shops and there was no bread; I would be unlikely ever to say that I went to the shops and bread did not exist there.

You also have to be bear in mind that certain quantities can be arbitrarily set to zero. For example, the gravitational potential energy associated with an object varies with height above the Earth, and can be defined on a scale with an arbitrary origin, so that the potential energy might take a zero value but would still exist.

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