Final answer:
To prove ∀x(P(x)→Q(x))⊨∀xP(x)→∀xQ(x), we can utilize the logical form known as a disjunctive syllogism in valid deductive inferences. This entails showing that if ∀x(P(x)→Q(x)) is true, then ∀xP(x)→∀xQ(x) is also true.
Step-by-step explanation:
In order to prove the logical statement ∀x(P(x)→Q(x))⊨∀xP(x)→∀xQ(x), we have to understand the concept of a disjunctive syllogism. This is a common argument form in the branch of logic known as valid deductive inferences.
A disjunctive syllogism is an argument where, if one particular statement is true, there is a corresponding conclusion that also must be true. So, for our problem: if ∀x(P(x)→Q(x)) is a true statement that is, for all x, if P(x) is true, then Q(x) is also true, then the corresponding conclusion wants us to show if ∀xP(x) is true then ∀xQ(x) is true. In this case, both the premises and the conclusion is logically correct, and therefore the statement ∀x(P(x)→Q(x))⊨∀xP(x)→∀xQ(x) is proved.
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