Incorrect Answer.
The semantic entailment
x P(x) 
x Q(x) 
x (P(x) Q(x))
is valid in predicate logic if, and only if all models M (that have
relations PM and QM defined) that satisfy
x P(x) x Q(x)
also satisfy
x (P(x) Q(x)).
- A model M satisfies x P(x) x Q(x)
if, and only if the sets PM and QM are both non-empty
(= contain at least one element).
- A model M satisfies x (P(x) Q(x))
if, and only if the intersection of the sets PM
and QM is non-empty.
With these insights at hand, it is not hard to come up with a
counterexample. Essentially, we only need to construct
non-empty sets PM and QM such that their
intersection is empty.
So let the model M be given
by
A = {a,b};
PM = {a}
QM =
{b}.
Please verify that we have
M x P(x) x Q(x), but that we do not have
M
x (P(x) Q(x)). Thus, the semantic entailment
above is not valid in predicate logic.
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