Editing Openai/696b7f8a-9760-8006-a1b5-89ffd7c5d2d9 (section)

=== Assistant: Conclusion / TL;DR: No — your biconditional analysis ===
Conclusion / TL;DR: No — your biconditional analysis

∀x [¬(x=m)  ⟺  L(x,m)]\forall x\,[\neg(x=m)\iff L(x,m)]∀x[¬(x=m)⟺L(x,m)]
is not correct for the intended meaning of “Everybody likes Mary, except Mary herself.” The standard and correct logical form is

∀x [¬(x=m)→L(x,m)],\forall x\,[\neg(x=m)\to L(x,m)],∀x[¬(x=m)→L(x,m)],
and if you really want to force Mary not to like herself, you’d add ¬L(m,m)\neg L(m,m)¬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. Mathematics Stack Exchange<ref>{{cite web|title=Mathematics Stack Exchange|url=https://math.stackexchange.com/questions/900721/analyzing-logical-forms-involving-quantifiers|publisher=Mathematics Stack Exchange|access-date=2026-01-18}}</ref>

Example:
Say the domain is people {m,a,b}\{m,a,b\}{m,a,b} where mmm is Mary, and define
* L(a,m)=L(a,m)=L(a,m)= true and L(b,m)=L(b,m)=L(b,m)= true (everyone else likes Mary).
* L(m,m)=L(m,m)=L(m,m)= false (Mary does not like herself). Then ∀x[¬(x=m)→L(x,m)]\forall x[\neg(x=m)\to L(x,m)]∀x[¬(x=m)→L(x,m)] is true and matches the intended meaning, but your biconditional ∀x[¬(x=m)  ⟺  L(x,m)]\forall x[\neg(x=m)\iff L(x,m)]∀x[¬(x=m)⟺L(x,m)] is false because e.g. for aaa, L(a,m)L(a,m)L(a,m) is true but ¬(a=m)\neg(a=m)¬(a=m) is also true — making the biconditional true — while for Mary mmm, ¬(m=m)\neg(m=m)¬(m=m) is false but L(m,m)L(m,m)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. Mathematics Stack Exchange<ref>{{cite web|title=Mathematics Stack Exchange|url=https://math.stackexchange.com/questions/4972729/conditional-or-biconditional-for-except|publisher=Mathematics Stack Exchange|access-date=2026-01-18}}</ref>

Why the conditional is correct:
The English “except” here functions like a restrictive clause: “for every person xxx, if xxx is not Mary, then xxx likes Mary.” That’s exactly what the conditional ¬(x=m)→L(x,m)\neg(x=m)\to L(x,m)¬(x=m)→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. Mathematics Stack Exchange<ref>{{cite web|title=Mathematics Stack Exchange|url=https://math.stackexchange.com/questions/900721/analyzing-logical-forms-involving-quantifiers|publisher=Mathematics Stack Exchange|access-date=2026-01-18}}</ref>

Why the biconditional is too strong:
A biconditional P  ⟺  QP\iff QP⟺Q means P→QP\to QP→Q and Q→PQ\to PQ→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. Mathematics Stack Exchange<ref>{{cite web|title=Mathematics Stack Exchange|url=https://math.stackexchange.com/questions/4972729/conditional-or-biconditional-for-except|publisher=Mathematics Stack Exchange|access-date=2026-01-18}}</ref>

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:

(∀x [¬(x=m)→L(x,m)])  ∧  ¬L(m,m),\big(\forall x\,[\neg(x=m)\to L(x,m)]\big)\;\land\;\neg L(m,m),(∀x[¬(x=m)→L(x,m)])∧¬L(m,m),
making both the general case about others and the exception about Mary explicit. Mathematics Stack Exchange<ref>{{cite web|title=Mathematics Stack Exchange|url=https://math.stackexchange.com/questions/4972729/conditional-or-biconditional-for-except|publisher=Mathematics Stack Exchange|access-date=2026-01-18}}</ref>

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.