How would a statement beginning, "For all $x$ in set $S$ except $x=a$..." be translated into symbolic predicate logic? I'm somewhat of a purist in symbols and am less than satisfied with "$\forall x \in S$ except $x = a$..." or "$\forall x \in S$ where $x \ne a$...".
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The exact correspondence in first order language is as follows:
$∀x((x∈S ∧ x≠a) → ⋯)$
It reads: for all $x$, if $x$ belongs to $S$ and $x$ is not $a$, then....
$x≠a$ is just a short form for $￢(x=a)$
I would use $(\exists a\in S)(\forall x\in S)(x\neq a \Leftrightarrow \cdots)$. It's not a direct correspondence to the English, but not everything in symbolic logic is.