There is another quantifier, besides existential and universal: the *unique existential quantifier*.

**Definition.** The sentence is true if there is exactly one in the universe so that is true. We read There is a unique such that ."

Notice that mathematically, *unique* means something rather different from what it means in ordinary English. Mathematically, *unique* always appears attached to some property. If we said Maxine's a unique person, mathematically we'd mean There is only one person in all the universe, and that one person is Maxine.

**Proposition**. is equivalent to each of the following:

**Proof.**

First we'll show that each of (1) and (2) is enough to guarantee . (1) guarantees that there is at least one with . If we had two *distinct* members of the universe, say, and , satisfying and , then we'd have , so and wouldn't actually be distinct after all. Thus there is at most one with .

Now assume (2) is true. We know that there is at least one with . Suppose we had another, say . Then applying with , we see that . So was to begin with. So there is at most one with .

Now we'll show that guarantees (1) and (2).

First, examine (1). Since we are proving a statement of the form , we must establish both conjuncts and . The first conjunct is clearly true; there is a unique with , therefore there is some with . Now let's prove the second conjunct, . If this were false, we'd have . Thus we have two distinct values of for which is true. But this contradicts .

Now consider (2). We need to find the special ; let's use the given by . Such an has . Now given with , we see that since there is exactly one value with , and and , it must have been that .

Thus we've shown that guarantees (1) and (2) and each of (1) and (2) guarantees . This completes the proof.

We can describe (1) above as saying: there is at least one with , and there is at most one with ).

We can describe (2) as saying: there is a special with has both , and the property that whenever is true, must be the same as .

Here we could say some words about how to prove a statement of the form , but we'll postpone that a bit.

Exercising our powers of **what-if-not thinking**, let's ask how could be false. Consider the following statements:

- There is a unique President of the United States.
- There is a unique US Senator from Florida.
- There is a unique US Senator from Washington, DC.

(1) is true. (2) and (3) are false, but false for different reasons. Let's see why:

**Exercise**. Compute the denial of , using the fact that can be expressed

Explain the relevance of this to the problem of uniqueness of Senators from Florida and from Washington, DC.