To determine whether the argument given below is VALID or INVALID:
[tex]\[
\begin{array}{l}
q \rightarrow p \\
p \rightarrow r \\
\therefore q \rightarrow r
\end{array}
\][/tex]
we'll analyze the logical implications step-by-step:
1. Understanding the Given Statements:
- [tex]\(q \rightarrow p\)[/tex]: This means if [tex]\(q\)[/tex] is true, then [tex]\(p\)[/tex] must be true.
- [tex]\(p \rightarrow r\)[/tex]: This means if [tex]\(p\)[/tex] is true, then [tex]\(r\)[/tex] must be true.
2. Construct the Argument Structure:
- We need to determine if [tex]\(q \rightarrow r\)[/tex] logically follows from the given premises [tex]\(q \rightarrow p\)[/tex] and [tex]\(p \rightarrow r\)[/tex].
3. Constructing the Proof:
- Assume [tex]\(q\)[/tex] is true.
- From the first premise [tex]\(q \rightarrow p\)[/tex], if [tex]\(q\)[/tex] is true, then [tex]\(p\)[/tex] is true.
- Now, [tex]\(p\)[/tex] being true satisfies the second premise [tex]\(p \rightarrow r\)[/tex]. Since [tex]\(p\)[/tex] is true, [tex]\(r\)[/tex] must also be true.
4. Reasoning to the Conclusion:
- Therefore, assuming [tex]\(q\)[/tex] is true leads to [tex]\(r\)[/tex] being true through [tex]\(p\)[/tex], showing that [tex]\(q \rightarrow r\)[/tex] holds.
- Thus, [tex]\(q \rightarrow r\)[/tex] follows logically from the premises.
Based on the above logical reasoning:
[tex]\[
\therefore q \rightarrow r \text{ is a VALID argument.}
\][/tex]
