# Proposition: Al knows only Bill

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))

?

• If AL does not know Bill then the implication is still true – David Peterson Oct 17 at 4:58

Because

∀x(K(Al, x) → (x = Bill))

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

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.

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 $$K(\text{Al}, x) \text{ iff } (x = \text{Bill})$$.

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:

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