Proposition: Al knows only Bill

Question

This is an example from my textbook:

Translate the proposition "Al knows only Bill" into symbolic form. Let’s use K(x, y) for the predicate x knows y. The translation would be K(Al, Bill) ∧ ∀x (K(Al, x) → (x = Bill)).

Why do we need:

K(Al, Bill)

isn't this enough:

∀x(K(Al, x) → (x = Bill))

?

Answer 1

Because

∀x(K(Al, x) → (x = Bill))

also allows the possibilty that Al doesn't know any x.

Answer 2

It is an implication. If Al knows $x$, then $x$ is Bill. But without the first part we do not actually know if Al knows Bill, just that if Al does know somebody, it must be Bill.

Answer 3

If the proposition $Q$ is true, then $P \rightarrow Q$ is true regardless of whether $P$ is true. In particular, $K(\text{Al}, \text{Bill}) \rightarrow (\text{Bill} = \text{Bill})$ is true regardless of whether Al knows Bill. You need $K(\text{Al}, x) \text{ iff } (x = \text{Bill})$.

Answer 4

It is implied that Al knows Bill, and $\forall x(K(Al, x) \to (x = Bill))$ would also be true if Al does not know Bill

So, you either need to explicitly add $K(Al, Bill)$, or do:

$$\forall x (K(Al, x) \leftrightarrow (x = Bill))$$