Choose the correct choice(s) regarding the following proportional logic…
2021
Choose the correct choice(s) regarding the following proportional logic assertion \(S\):
\(S: (( P \wedge Q) \rightarrow R) \rightarrow (( P \wedge Q) \rightarrow (Q \rightarrow R))\)
- A.
\(S\)is neither a tautology nor a contradiction - B.
\(S\)is a tautology. - C.
\(S\)is a contradiction. - D.
The antecedent of
\(S\)is logically equivalent to the consequent of\(S\).
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Correct answer: B, D
Key insight: reduce the formula by setting A = P ∧ Q.
Let A = P ∧ Q. Then S becomes (A → R) → (A → (Q → R)).
Case A is false: both A → R and A → (Q → R) are true, so the implication is true.
Case A is true: then Q is true, so Q → R has the same truth value as R. Hence A → (Q → R) has the same truth value as A → R, so the whole formula has the form X → X, which is true.
Conclusion: The formula S is true in every valuation (a tautology). Equivalently, the antecedent (P ∧ Q) → R is logically equivalent to the consequent (P ∧ Q) → (Q → R).
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