Final answer:
The WFF in question is a valid argument since it follows that if the premise (∀x)p(x) guarantees (∀x)q(x), then universally, p(x)→q(x) must be true, and a counterexample cannot be generated to show otherwise.The correct answer is option (a)
Step-by-step explanation:
The student's question concerns the validity of the well-formed formula (WFF): (∀x)p(x)→(∀x)q(x)→(∀x)[p(x)→q(x)]. To assess the validity of this argument, we must explore whether its structure guarantees that if the premises are true, the conclusion cannot be false. In this case, the premises assert that for all x, if p(x) is true, then q(x) is also true and subsequently if q(x) is true, then it must follow that for all x, if p(x) is true, then q(x) is true.
Valid deductive inferences, such as the disjunctive syllogism, are foundational in assessing the soundness of arguments. Utilizing the concepts of sufficient and necessary conditions to analyze logical statements, we find that if the presence of condition p(x) guarantees q(x), then the argument's form is valid. This is because satisfying the initial condition (p(x)) ensures the secondary condition (q(x)).
To prove invalidity, a counterexample must show that the premises can be true while the conclusion is false. However, in the structure of this particular argument, if for every x the premise that p(x) leads to q(x) holds, then universally (for all x), p(x) impelling q(x) would also hold, making the argument valid. Therefore, based on logical standards, the argument presented is valid.