How can I translate the following statement into Predicate logic and what will be the domain? Basically I know how to find domain and predicate logic.
Example: All americans eat cheese burger
for this, First domain will be
Americans and logic will be A(x):x eat cheese burger. So, ∀x L(x). Second domain will be people in america and logic will be P(x):x people in america. A(x):x eat cheese burger. So, ∀A [P(x)->A(x)] . But I am confused about the following one. Since, it's a phrase that's why I don't understand that what will be the domains and logic.
Every cloud has a silver lining
Your first analysis is pretty good:
define:
$A(x)$: x eats cheese burgers
$P(x)$: x is an American
If you assume the domain is all Americans, then the statement 'all Americans eat cheese burgers' becomes $\forall x A(x)$
But if the domain is larger than that (say: all people in the world, or all objects in the world), then you have to use $\forall x (P(x) \rightarrow A(x))$
For the second one, you have a similar choice:
The domain is all clouds
The domain is larger than all clouds (so again: maybe 'all object in the world')
Using:
$C(x)$: x is a cloud
$S(x)$: $x$ has a silver lining
You get $\forall x \: S(x)$ if you use the first option
And $\forall x (C(x) \rightarrow S(x))$ is you use the second option