Alex likes anything that contains chocolate.
a - Alex
L(x,y) - x likes y
C(x) - x contains chocolate
$1. \forall x \space (C(x) \implies L(a,x)) $
$2. \forall x \space (C(x) \space \text{^} \space L(a,x)) $
Is there a difference between 1 and 2? I know that the truth tables are different. Is one an incorrect representation in predicate logic?
Edit:
I've removed the truth table as it was completely bogus for predicate logic. I'm still confused with using the $\forall$ vs $\exists$ with an implication. Would $\exists x \space (C(x) \space \text{^} \space L(a,x)) $ be a correct representation of "There exists something that contains chocolate and Alex likes it"? Is this equivalent to "Alex likes anything that contains chocolate"?
Proposition #1 is the correct representation of the English sentence. In literal terms it says that "For all objects, if it contains chocolate, then Alex likes it."
Another way of stating #1 symbolically is $\forall x \neg(C(x) \land \neg L(a, x))$ which is equivalent by deMorgan's Laws to $\forall x (\neg C(x) \lor L(a, x))$, which translates to "For any object, either it contains chocolate or Alex likes it."
Proposition #2 does not represent the English statement correctly, saying that "For all objects, it contains chocolate, and Alex likes it." This is clearly false since not all objects contain chocolate (although that would be nice).