I'm currently reading about why one should avoid using implication in an existentially quantified sentence.
I'm having a hard time understanding why the following
$$\exists x \left( \mbox{Student} (x) \land \mbox{At}(x, \text{Stanford}) \implies \mbox{Smart}(x) \right)$$
means "Someone at Stanford is smart OR someone does not go to Stanford"
I've used implication elimination and distributivity to simplify the previous sentence into:
$$\exists x \neg Student(x) \lor \neg At(x, Stanford) \lor Smart(x)$$
To me, this sentences means "There is a person who is not a student or not at Stanford or they're smart."
What am I missing in interpreting the previous sentence as "Someone at Stanford is smart OR someone does not go to Stanford"?
Assuming all the x in the sentence are bound by ∃x, you have for some x, if (x is a student & x is at Stanford) then (x is smart). In predicate logic, we use the Material Implication interpretation of a conditional : A⟹B is logically equivalent to (~A v B).
On this interpretation, sentences such as "If the moon is made of green cheese then I'm 20 feet tall" come out true. It may not seem intuitive, but this interpretation has desirable properties. Basically any conditional is true as long as it's not the case that the antecedent A is true and the consequent B is false.
In your example, if we define C:=(x is a student) & D:=( x is at Stanford), and B:=(x is smart), we have (C&D)⟹B. So by the material implication interpretation of conditionals, we have either (C&D) is false or B is true. [And if (C&D) is not true, then either C or D (or both) is false, i.e. (~C v ~D).]