I have been solving the following problem from How to Prove book:
Analyze the logical forms of the following statement:
Everyone likes Mary, except Mary herself.
Now, the above sentence conveys the following meaning (to me): Everyone likes Mary but she doesn't like herself.
And I have translated it into the following logical forms:
L(x,y) = x likes y
∀x L(x,m) ∧ ¬L(m,m) (m refers to Mary)
But the answer is something like this:
∀x(¬(x = m) → L(x, m)), where L(x, y) stands for “x likes y,” and
m stands for Mary.
How does implication come here ? What is the thought process involved here ?
If $\forall x, x\ne m\Rightarrow L(x,m)$.
The only person who doesn't like Mary is Mary herself. So if someone doesn't like Mary, that someone must be Mary. Or the contrapositive; if someone isn't Mary, that someone likes Mary.
In your formulation, you have Mary liking Mary ($\forall x L(x,m)$) and Mary not liking Mary ($\neg L(m,m)$). These two conjuncted form a contradiction.