Consider the following logical statement:[math: (P⇒Q)∧¬Q](P \Rightarrow Q) \land
eg QWhich of the following is true about this statement?

Question

Consider the following logical statement:[math: (P⇒Q)∧¬Q](P \Rightarrow Q) \land 
eg QWhich of the following is true about this statement?

BlackTom AI · Accepted Answer

Take a moment to rewrite the given statement in a form that's easier to analyze. The statement is (P⇒Q) ∧ ¬Q. Recall that P⇒Q is equivalent to ¬P ∨ Q. So the whole expression becomes (¬P ∨ Q) ∧ ¬Q.

Option a: It is a tautology. A tautology is a statement that is true for every possible truth assignment of P and Q. Here, if we set P = false and Q = false, (¬P ∨ Q) becomes (true ∨ false) which is true, and ¬Q is true, so the conjunction is true in that case. But to be a tautology, it must be true for all combinations, including P = true and Q = true where (¬P ∨ Q) is (false ∨ true) = true and ¬Q is false, making the conjunction false. Therefore, the statement is not true in all cases, so it is not a tautology.

Option b: It is a contradiction. A contradiction is true in no possible interpretation. From the previous reduction, (¬P ∨ Q) ∧ ¬Q simplifies to (¬P ∧ ¬Q) since (Q ∧ ¬Q) is always false and does not contribute any true case. The pair (¬P ∧ ¬Q) is true whenever both P and Q are false, which is a valid assignment (P = false, Q = false). Hence, the statement can be true, so it is not a contradiction.

Option c: It implies that P is false. Examining the simplified form (¬P ∧ ¬Q), every truth assignment that makes the original statement true must satisfy ¬P, i.e., P is false. Therefore, the statement indeed implies that P is false. This is a correct reading of the logical consequence of the conjunction with ¬Q.

Option d: It implies that P is true. This would require P to be true in all cases where the statement holds. But we just observed that the statement holds only when P is false (since ¬P must be true as part of the conjunction). Therefore, the statement does not imply P is true, and this option is incorrect.

Consider the following logical statement:[math: (P⇒Q)∧¬Q](P \Rightarrow Q) \land \neg QWhich of the following is true about this statement?

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