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:
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.