Conclusion / TL;DR: No — your biconditional analysis
\forall x\,[\neg(x=m)\iff L(x,m)]
is not correct for the intended meaning of “Everybody likes Mary, except Mary herself.” The standard and correct logical form is
\forall x\,[\neg(x=m)\to L(x,m)],
and if you really want to force Mary not to like herself, you’d add \neg L(m,m). The biconditional is too strong because it says “liking Mary” is equivalent to not being Mary. That’s not what the English sentence logically entails.
Example:
Say the domain is people \{m,a,b\} where m is Mary, and define
- L(a,m)= true and L(b,m)= true (everyone else likes Mary).
- L(m,m)= false (Mary does not like herself).
Then \forall x[\neg(x=m)\to L(x,m)] is true and matches the intended meaning, but your biconditional \forall x[\neg(x=m)\iff L(x,m)] is false because e.g. for a, L(a,m) is true but \neg(a=m) is also true — making the biconditional true — while for Mary m, \neg(m=m) is false but L(m,m) is also false, which makes the biconditional true again, yet this equivalence says only non-Marys like Mary and Mary never likes herself — a much stronger claim than intended.
Why the conditional is correct:
The English “except” here functions like a restrictive clause: “for every person x, if x is not Mary, then x likes Mary.” That’s exactly what the conditional \neg(x=m)\to L(x,m) captures: whenever you take someone who isn’t Mary, they like Mary. It doesn’t force anything about Mary herself — which is fine if the problem only wants to say that others like Mary.
Why the biconditional is too strong:
A biconditional P\iff Q means P\to Q and Q\to P. For your formula, that means
- if someone is not Mary, then they like Mary, and
- if someone likes Mary, then they are not Mary.
That second part rules out the possibility of Mary liking herself — including any context where she might — and it also imposes the idea that only non-Marys can like her. Neither of these follow logically from just saying “everybody else likes Mary.” Most translations of that English sentence don’t logically entail these stronger requirements.
Optional refinement:
If the exercise really wants to assert both that everyone except Mary likes Mary and that Mary does not like herself, the cleanest way to express it in predicate logic is:
\big(\forall x\,[\neg(x=m)\to L(x,m)]\big)\;\land\;\neg L(m,m),
making both the general case about others and the exception about Mary explicit.
If you want a detailed truth-table breakdown showing why implication \to differs from biconditional \iff in this context, I can walk through that too.