Let's analyze each argument in terms of the propositions and rule of inference used.

Argument 1:

Statement: If $n$ is a real number such that $n > 3$, then $2n > 6$. Suppose that $n \geq 2$, can you conclude $2n > 6$?

Propositions:

• $P$: $n > 3$
• $Q$: $2n > 6$
• Premise: $P \rightarrow Q$
• Additional assumption: $n \geq 2$

Analysis:

• The premise is the implication $P \rightarrow Q$, meaning if $n > 3$, then $2n > 6$.
• However, the assumption you are given is $n \geq 2$, which includes values of $n$ such as 2, 2.5, and 3 that do not satisfy $n > 3$.
• Since the premise only applies if $n > 3$, and the new assumption doesn't guarantee $n > 3$, you cannot conclude $2n > 6$. The condition for the premise to apply isn't met under the new assumption.

Conclusion:

• Reason for no conclusion: The assumption $n \geq 2$ does not imply $n > 3$, so you cannot apply the premise $P \rightarrow Q$.
• Rule of inference: No valid inference rule can be applied here due to the invalid assumption.

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Argument 2:

Statement: If $n$ is a real number such that $n > 2$, then $n^2 > 4$. Suppose that $n^2 \leq 4$, then $n \leq 2$.

Propositions:

• $P$: $n > 2$
• $Q$: $n^2 > 4$
• Premise: $P \rightarrow Q$
• New assumption: $n^2 \leq 4$

Analysis:

• The premise states $P \rightarrow Q$, meaning if $n > 2$, then $n^2 > 4$.
• The contrapositive of $P \rightarrow Q$ is $\neg Q \rightarrow \neg P$, i.e., if $n^2 \leq 4$, then $n \leq 2$.
• Since the assumption is $n^2 \leq 4$, you can apply the contrapositive of the original statement to conclude that $n \leq 2$.

Conclusion:

• Valid conclusion: Using the contrapositive of $P \rightarrow Q$, you can conclude that $n \leq 2$ when $n^2 \leq 4$.
• Rule of inference: Contrapositive rule.

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Summary:

1. First argument:

   • Propositions: $P \rightarrow Q$ where $P$: $n > 3$, $Q$: $2n > 6$, but the assumption $n \geq 2$ does not satisfy $P$, so no conclusion.
   • Reason: Invalid assumption.
   • Inference rule: None applicable.

2. Second argument:

   • Propositions: $P \rightarrow Q$ where $P$: $n > 2$, $Q$: $n^2 > 4$, and the contrapositive $\neg Q \rightarrow \neg P$ applies.
   • Conclusion: $n \leq 2$ (contrapositive rule).

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Do you want further details or have any questions?

Related Questions:

1. What is the difference between a contrapositive and converse in logic?
2. Can you provide an example of Modus Ponens?
3. How do you recognize when a rule of inference cannot be applied?
4. What is the contrapositive of $P \rightarrow Q$?
5. How can you formally prove the contrapositive inference rule?

Tip:

Always check whether the premise or the contrapositive of a logical statement applies when given assumptions that seem to reverse the implication.