Final answer:
The argument A ∧ (B → A) → B′ can be proven using modus tollens in propositional logic. This analysis shows that if the premises are true, it logically follows that the conclusion must also be true, thereby confirming the argument's validity.
Step-by-step explanation:
Using propositional logic to prove the validity of the argument A ∧ (B → A) → B′ involves analyzing the logical structure of the argument. First, we recognize that the argument is a conditional statement and thus can be associated with the valid inference form known as modus tollens. In this form, if the antecedent is true, and the consequent is false, then the negation of the premise leading to the antecedent must be true.
Let's break down the argument:
∀: A (Statement A is true)
∀: B → A (If B then A)
∀: A ∧ (B → A) (Both statement A and the conditional 'if B then A' are true)
∀: B′ (Not B must be true if the previous statements are true)
We perform a logical analysis to establish the truth of the conclusion based on the premises. If the conjunction A ∧ (B → A) is true, it implies A is true. As B → A is true, and we know A is true, the only way this can be the case is if B is false, because if B were true, then the conditional would not affect the truth of A. Therefore, B must be false (B′). This deduction shows that the argument follows a valid logical form and the conclusion necessarily follows from the premises.