Incorrect Answer.
The semantic entailment
x (P(x) 
Q(x)) 
x P(x) 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) Q(x)) also satisfy
x P(x) x Q(x).
- A model M satisfies x (P(x) Q(x)) if, and only if the union of the sets PM and QM equals the model's set of values, A.
- A model M satisfies x P(x) x Q(x)
if, and only if the set PM or the set QMequals the model's set of values, A.
With these insights at hand, it is not hard to come up with a counterexample. Let the model M be given by
A = {a,b};
PM = {a}
QM = {b}.
Please verify that we have
M x (P(x) Q(x)), but that we do not have
M
x P(x) x Q(x). Thus, the semantic entailment
above is not valid in predicate logic.
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