# Help with English to Predicate Logic

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"?

• The second one says in particular $\forall xC(x)$. Can you see why this isn't right? – Git Gud Dec 9 '14 at 1:25
• Your truth table isn't correct for doing predicate logic. The statement C(x) (or L(a,x)) is not truth-functional, ie it is not true or false absolutely. To get statements which ARE true or false you first have to define a universe, say the set {a,b} and THEN consider whether statements like C(a) and C(b) are true or false. Just think about the English statement "x contains chocolate." Is that true or false? Depends on what 'x' is! – Kevin Driscoll Dec 9 '14 at 1:39
• Ah! I knew it. While I was drawing out the table I had a nagging feeling that this was all wrong. Lets forget about the table. I'm still confused with the 2 expressions. Do all predicate logic statements need an implication? – Zee Dec 9 '14 at 1:43

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."