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 its set of values, A, equals the set QM
whenever A equals the set PM.
- A model M satisfies x (P(x) Q(x))
if, and only if the set PM is a subset of QM.
With these insights at hand, it is not hard to come up with a
counterexample. Essentially, we only need to ensure
that PM is not a subset of QM
and that the implication x
P(x) x Q(x)) is satisfied. But the
latter we can simply achieve by choosing a model that does not satisfy
the premise of the implication! 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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