IDNLearn.com: Your trusted source for finding accurate and reliable answers. Join our knowledgeable community to find the answers you need for any topic or issue.

Determine if the following are VALID or INVALID:

[tex]\[
\begin{array}{l}
q \rightarrow p \\
p \rightarrow r \\
\therefore q \rightarrow r
\end{array}
\][/tex]


Sagot :

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]