\( \exists x P \equiv \neg \forall x \neg P \) convert this sentence into unquantified sentence: \( P P \vee Q \equiv \neg(\neg P \wedge \neg Q) \) \( P \vee Q \equiv \neg(P \wedge \neg Q) \) \( P P \wedge Q \equiv \neg(P \vee \neg Q) \) \( P A \wedge Q \neg \neg(\neg P \vee \neg Q) \)

To convert the given quantified sentences into unquantified sentences, we will interpret the logical expressions without the quantifiers. 1. **For the expression** \( \exists x P \equiv \neg \forall x \neg P \): - This means "There exists an \( x \) such that \( P \) is true" is equivalent to "It is not the case that for all \( x \), \( P \) is false." - In unquantified terms, we can express this as: "At least one instance of \( P \) is true." 2. **For the expression** \( P \vee Q \equiv \neg(\neg P \wedge \neg Q) \): - This is already in an unquantified form. It states that "Either \( P \) is true or \( Q \) is true" is equivalent to "It is not the case that both \( P \) and \( Q \) are false." - In unquantified terms, we can say: "At least one of \( P \) or \( Q \) is true." 3. **For the expression** \( P \wedge Q \equiv \neg(P \vee \neg Q) \): - This states that "Both \( P \) and \( Q \) are true" is equivalent to "It is not the case that \( P \) is true or \( Q \) is false." - In unquantified terms, we can say: "Both \( P \) and \( Q \) are true." 4. **For the expression** \( P \wedge Q \equiv \neg \neg(\neg P \vee \neg Q) \): - This states that "Both \( P \) and \( Q \) are true" is equivalent to "It is not the case that either \( P \) is false or \( Q \) is false." - In unquantified terms, we can say: "Both \( P \) and \( Q \) are true." In summary, the unquantified interpretations of the logical expressions are: 1. "At least one instance of \( P \) is true." 2. "At least one of \( P \) or \( Q \) is true." 3. "Both \( P \) and \( Q \) are true." 4. "Both \( P \) and \( Q \) are true."