# When to use AND and when to use Implies in first-order logic?

I am trying to learn the theory behind first-order logic (FOL) and do some practice runs of converting statements into the form of FOL.

One issue I keep running into is hesitating on whether to use an AND ($$\land$$) statement or an IMPLIES ($$\rightarrow$$) statement.

I have seen examples such as "Some boys are intelligent" turned into:

$$\exists x \text{boys}(x) \land \text{intelligent}(x)$$

Can I make a general assumption that when I see $$x$$ is/are $$y$$, I can use an AND?

With a statement such as "All movies directed by M. Knight Shamalan have a supernatural character", I feel that that statement can be translated to either:

$$\forall x, \exists y \; \text{directed}(\text{Shamalan}, x) \rightarrow \text{super-natural character}(y)$$

or

$$\forall x, \exists y \; \text{directed}(\text{Shamalan}, x) \land \text{super-natural character}(y)$$

Is there a better way to distinguish between when to use one or the other?

• i don't think ∃x boys(x) ∧ intelligent(x) is correct for the statement "Some boys are intelligent". there might be some boys who aren't intelligent, for which the predicate fails. Jun 18 '20 at 19:53
• @nikhilbalwani What would be a clearer more concise way of writing out that statement in terms of first order logic?
– ndrb
Jun 18 '20 at 20:17