Question

Which of the following logical statements is not valid?

• A. \( \forall x\, P(x) \Rightarrow \exists x\, \neg P(x) \)
• B. \( \forall x\, P(x) \Rightarrow \exists x\, P(x) \)
• C. \( \exists x\, P(x) \Rightarrow \forall x\, P(x) \)
• D. \( \exists x\, P(x) \Leftrightarrow \forall x\, P(x) \)

Hint:

Remember: Universal truth implies existence, but existence does not imply universal truth.

Answer 1

The Correct Option is A

Step 1: Understanding the Topic
This question tests the understanding of predicate logic, specifically the relationships between statements using the universal quantifier ($\forall$, "for all") and the existential quantifier ($\exists$, "there exists"). A logical statement (implication, equivalence) is valid if it is true for any possible predicate $P(x)$ and any non-empty domain of discourse.
Step 2: Key Approach - Evaluating Each Statement
We will analyze each logical statement to see if it holds true under all circumstances or if we can find a counterexample that makes it false.
Step 3: Detailed Explanation
Let's consider a domain of discourse, for example, the set of all integers.

(A) \( \forall x\, P(x) \Rightarrow \exists x\, \neg P(x) \): This statement reads, "If $P(x)$ is true for all $x$, then there exists an $x$ for which $P(x)$ is false." This is a direct contradiction. If the premise ($\forall x\, P(x)$) is true, it means there are NO elements for which $P(x)$ is false. Therefore, the conclusion ($\exists x\, \neg P(x)$) must be false. An implication `True $\Rightarrow$ False` is logically false. Thus, this statement is not valid.
(B) \( \forall x\, P(x) \Rightarrow \exists x\, P(x) \): This statement reads, "If $P(x)$ is true for all $x$, then there exists at least one $x$ for which $P(x)$ is true." This is always true for any non-empty domain. If a property holds for every element, it must hold for at least one of them. This is valid.
(C) \( \exists x\, P(x) \Rightarrow \forall x\, P(x) \): This statement reads, "If there exists an $x$ for which $P(x)$ is true, then $P(x)$ must be true for all $x$." This is not necessarily true. For example, let $P(x)$ be "x is an even number". There exists an even number (e.g., 2), but not all numbers are even. So, this is not valid.
(D) \( \exists x\, P(x) \Leftrightarrow \forall x\, P(x) \): This biconditional is false because the implication from right-to-left is valid (B), but the implication from left-to-right is not valid (C). So this is not valid.
The question asks for the statement that is "not valid". Statements (A), (C), and (D) are all not valid in general. However, statement (A) is a fundamental logical contradiction, making it the "most" invalid or the intended answer representing an impossible logical step.
Step 4: Final Answer
Statement (A) is a logical self-contradiction and therefore is fundamentally not valid.