Textbook
The assertion $\exists x\in A:P(x)$ says that at least one element of domain $A$ makes predicate $P(x)$ true. Such an element is a witness. The symbol $\exists!$ strengthens the requirement: a qualifying element exists and is unique. Ordinary $\exists$ permits multiple witnesses.
Existence over an empty domain is false because no witness can be supplied. Negation gives $\neg(\exists x\in A:P(x))\Leftrightarrow\forall x\in A:\neg P(x)$. Refuting existence requires ruling out every allowed candidate.
Scope determines when the witness is chosen. The assertion $\exists y\in\mathbb R\;\forall x\in\mathbb R:y>x$ requires one fixed $y$ and is false. It differs from allowing a new $y=x+1$ for each new $x$.
A witness must both belong to the declared domain and satisfy the predicate. A qualifying number outside that domain proves nothing about the assertion. Searching a finite domain can stop at the first success when existence alone is needed. In an infinite domain, finitely many failures do not establish nonexistence: a witness may remain unexamined. Refutation then needs a general argument covering every admissible candidate.
Definitions and further reading: Mathematics for Computer Science.
In plain language
The claim “there is a red sock in the drawer” is established by one red sock found there. A second does not undo that success and may avert a familiar household tragedy. The witness proves existence rather than uniqueness; “exactly one” also requires ruling out other qualifying candidates. A sock from another drawer does not qualify either, because the domain belongs to the claim itself.
Example
For the illustrative set $A=\{2,4,6\}$, the example seeks a value greater than 3. Four supplies a witness; six shows why a claim using $\exists!$ would be false.
The Existential Quantifier: One Witness Is Enough
Candidate 2
The first candidate fails, but the search is not finished.
Step 1 of 3: Candidate 2
Read all steps
Candidate 2
The first candidate fails, but the search is not finished.
- State
- 2excluded46
Witness 4
Four proves that a qualifying element exists.
- State
- 2excluded4included6
Second witness 6
Existence remains true; uniqueness is impossible.
- State
- 2excluded4included6included
The failure of two excludes only that candidate. Four already makes existence true; six is checked here to contrast existence with uniqueness. Restricting the domain to $\{2\}$ would make the assertion false, even though 4 still exists among the real numbers. The domain determines which witnesses are admissible. A good example from outside the domain is not a valid witness.
Related concepts: Predicates, The universal quantifier.