Understanding the Logical Equivalence Problem

The question asks us to identify which of the given options is logically equivalent to the set of premises provided. The premises are:

• Premise 1: $p \implies (q \lor r)$
• Premise 2: $q \implies s$
• Premise 3: $r \implies s$

We need to find an expression that is a logical consequence of these premises. The target equivalent expression is $p \implies s$. Let's demonstrate how to derive this conclusion from the premises.

Deriving the Logical Equivalence $p \implies s$

We can use the rules of logical inference to show that $p \implies s$ follows from the given premises.

1. Start with Premise 1: $p \implies (q \lor r)$. This statement means that if $p$ is true, then either $q$ is true, or $r$ is true, or both are true.
2. Consider the case where $p$ is true: If $p$ is true, then according to Premise 1, the disjunction $(q \lor r)$ must also be true.
3. Analyze the disjunction $(q \lor r)$: For $(q \lor r)$ to be true, at least one of $q$ or $r$ must be true. We can consider two cases:
   • Case A: $q$ is true. If $q$ is true, we can use Premise 2 ($q \implies s$). By the rule of Modus Ponens, if $q$ is true, then $s$ must be true.
   • Case B: $r$ is true. If $r$ is true, we can use Premise 3 ($r \implies s$). Again, by Modus Ponens, if $r$ is true, then $s$ must be true.
4. Combine the cases: In both possible scenarios arising from $p$ being true (either $q$ is true or $r$ is true), we conclude that $s$ must be true.
5. Conclusion: Since assuming $p$ is true leads necessarily to $s$ being true, we can conclude that $p \implies s$ is logically equivalent to the given set of premises.

This derivation shows that the implication $p \implies s$ correctly captures the logical consequence of the initial statements.

Analyzing the Options

Let's briefly look at why the other options are not necessarily equivalent:

• Option 1 ($q \implies r$): The premises do not provide enough information to conclude that $q$ implies $r$.
• Option 2 ($r \implies q$): Similarly, the premises do not guarantee that $r$ implies $q$.
• Option 4 ($s \implies p$): The premises establish that $p$ leads to $s$, not the other way around. It's possible for $s$ to be true while $p$ is false.

Therefore, the only option that is logically equivalent to the given set of premises is $p \implies s$.