Math Problem Statement

answer

Which of the following arguments is not a valid argument?

Solution

Let's go through each argument step by step to determine which one is not a valid argument.

Argument A:

  1. Premise 1: x(p(x)q(x))\forall x (p(x) \rightarrow q(x))

    • For every xx, if p(x)p(x) is true, then q(x)q(x) is true.
  2. Premise 2: xp(x)\exists x p(x)

    • There exists some xx such that p(x)p(x) is true.
  3. Conclusion: xq(x)\therefore \exists x q(x)

    • There exists some xx such that q(x)q(x) is true.

    Analysis: This argument is valid. If there exists an xx such that p(x)p(x) is true, and for every xx, p(x)p(x) implies q(x)q(x), then there must exist some xx such that q(x)q(x) is true.

Argument B:

  1. Premise 1: x(p(x)q(x))\exists x (p(x) \land q(x))

    • There exists some xx such that both p(x)p(x) and q(x)q(x) are true.
  2. Conclusion: xp(x)\therefore \exists x p(x)

    • There exists some xx such that p(x)p(x) is true.

    Analysis: This argument is valid. If there exists some xx such that both p(x)p(x) and q(x)q(x) are true, it directly implies that there exists some xx where p(x)p(x) is true.

Argument C:

  1. Premise 1: xp(x)\exists x p(x)

    • There exists some xx such that p(x)p(x) is true.
  2. Premise 2: xq(x)\exists x q(x)

    • There exists some xx such that q(x)q(x) is true.
  3. Conclusion: x(p(x)q(x))\therefore \exists x (p(x) \land q(x))

    • There exists some xx such that both p(x)p(x) and q(x)q(x) are true.

    Analysis: This argument is not valid. The existence of some xx such that p(x)p(x) is true and the existence of some (potentially different) xx such that q(x)q(x) is true does not guarantee the existence of a single xx where both p(x)p(x) and q(x)q(x) are true.

Argument D:

  1. Premise 1: x(p(x)q(x))\exists x (p(x) \rightarrow q(x))

    • There exists some xx such that if p(x)p(x) is true, then q(x)q(x) is true.
  2. Premise 2: p(a)p(a)

    • p(a)p(a) is true for some specific aa.
  3. Conclusion: q(a)\therefore q(a)

    • q(a)q(a) is true for the specific aa.

    Analysis: This argument is valid. If p(a)p(a) is true and there exists some xx (which could be aa) such that p(x)q(x)p(x) \rightarrow q(x), then q(a)q(a) must be true.

Conclusion:

The invalid argument is C. The premises do not logically lead to the conclusion.

Would you like a more detailed explanation or have any questions about this solution?

Here are 5 related questions that expand on this topic:

  1. What are the rules of inference used to determine the validity of logical arguments?
  2. How do existential quantifiers differ from universal quantifiers in logical arguments?
  3. Can you explain the difference between validity and soundness in a logical argument?
  4. How can we use counterexamples to demonstrate that an argument is not valid?
  5. How does the structure of an argument affect its validity?

Tip: When dealing with logical arguments, always ensure the conclusion follows directly from the premises by applying the correct rules of inference.