Predicate Logic Translating "All But One" I need to translate an English sentence including the phrase "all but one" into predicate logic. The sentence is: "All students but one have an internet connection." I'm not sure how to show "all but one" in logic.
I could say $\forall x ((x \neq a) \rightarrow I(x))$
$I(x)$ being "$x$ has an internet connection" 
But that clearly wouldn't work in this case, as we don't know which student it is. 
I could say that $\exists x(\neg I(x))$
But it doesn't seem like that has the same meaning. 
Thanks in advance for any help you can give!

"For all but one $\;x\;$, $\;P(x)\;$ holds" is the same as "there exists a unique $\;x\;$ such that $\;\lnot P(x)\;$ holds.
Normally the notation $\;\exists!\;$ is used for "there exists a unique" (just like $\;\exists\;$ is used for "there exists some").
If your answer is allowed to use $\;\exists!\;$, then the above gives you the answer.
If not, then there are different ways to write $\;\exists!\;$ in terms of $\;\exists\;$ and $\;\forall\;$.  The one I like best, which also results in the shortest formula, can be found in another answer of mine.
