# What does the statement "Xs exist" really mean?

If someone were to claim:

Xs exist.

what do they actually mean? Do they mean:

1. At least one X exists.

or

1. At least two Xs exist.

In other words, does the phrase "Unicorns exist" mean "at least one unicorn exists" or "at least two unicorns exist"?

• In logic "X's exist" means : (i) "At least one X exists"; i.e. the property X is instantiated. Thus, having not found until today any specimen of unicorn, we can say that "unicorns do not exist". Nov 7, 2014 at 10:37
• Sorry to be pedantic, but "X's exist" actually means "The exist owned by X" which does not make much sense. Nov 10, 2014 at 17:30
• Why at least two and not at least 3? Or N? Nov 10, 2014 at 17:32
• @AsphirDom because "X's" is a plural noun, so by a rule of English it refers to two or more X's. Nov 10, 2014 at 19:01
• @user132181: You are wrong. "X's" is possessive in English. "Xs" is a plural. Nov 10, 2014 at 19:23

According to Whitehead & Russell's Principia Mathematica, existence is a case of a propositional function being true of at least one value of the variable.

"There exists y such that φ(y)" is equivalent to "not all y such that φ(y) is false."

"There does not exist y such that φ(y)" is equivalent to "for all y, φ(y) is false."

It is important to understand that existence can only be asserted of a description.

We can say 'the author of Waverly exists' and we can say 'Scott is the author of Waverley', but 'Scott exists' is bad grammar. It can, at least, be interpreted as meaning, 'the person named "Scott" exists', but 'the person named "Scott"' is a description, not a name. Whenever a name is properly used as a name it is bad grammar to say 'that exists'.

If unicorn is interpreted as "the creature described in wikipedia-Unicorn", then

"Unicorns exist" means "there is at least one creature that fits the description given in wikipedia-Unicorn."

"Unicorns do not exist" means "Of all the things in the universe, none fits the description given in wikipedia-Unicorn."

Based on above definition, none-existence involves "all", and therefore is not so easy to prove as it first appears. In order to prove unicorns do not exist, one would have to examine every thing in the universe from the beginning til the end of time. Thus, as far as current human knowledge can warrant, whether unicorns exist or not remains unknown.

*Source: Russell, Bertrand. My Philosophical Development. Principia Mathematica: Philosophical Aspects. New York: Simon and Schuster, 1959

In all cases that I know of, this will mean (i). The statement "do X's exist?" is normally meant to mean the first-order logic existential quantifier, which is true if there is at least one X.

• Welcome to philosophy.se. This answer seems to agree with my sense in logic* or rather formal logic. And in that sense you're perfectly right. Nov 7, 2014 at 14:47

(i) At least one X exists

Ea:X(a)

(ii) At least two X's exist.

Ea,b:[X(a) & X(b) & a=/=b]

where X is a unary predicate. X(y) means y is an X.

Could also mean more than one X exists.

Which of the above two does "X's exist" mean?

I think most would accept the first interpretation. If you really meant the plural of X or more than one X, then the second.

• Doesn't answer my question. Nov 7, 2014 at 22:40

"X's exist" means "At least one x which belongs to X exists". Capital letters are mostly used for sets where small letters are usually used for elements. So, existence of a set is being non-empty, in other words, there is at least one element of that set.

Then, "unicorns exist" means "there is at least one unicorn in the universe".

• 'X' here is just a metavariable standing for the singular form of an English noun, not a set. What you've written in your answer is just a bunch of sophistry. Nov 10, 2014 at 13:12
• I got what you meant, but when you say "Unicorns exist", and write it formally, you say: X: Unicorns There exists an x which belongs to X. Nov 11, 2014 at 11:30