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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))

?

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    $\begingroup$ If AL does not know Bill then the implication is still true $\endgroup$ – David Peterson Oct 17 at 4:58
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Because

∀x(K(Al, x) → (x = Bill))

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

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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.

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

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