To prove that the premises:

1. $P \rightarrow Q$
2. $Q \rightarrow R$
3. $R \rightarrow S$
4. $S \rightarrow R$
5. $P \land S$

are inconsistent, we need to show that these premises together lead to a contradiction. Let's proceed step-by-step using formulas:

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Step 1: Start with $P \land S$

From the premise $P \land S$, we know:

• $P$ is true.
• $S$ is true.

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Step 2: Derive $Q$ from $P \rightarrow Q$

Since $P$ is true and $P \rightarrow Q$ is a premise, we can infer:

• $Q$ is true.

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Step 3: Derive $R$ from $Q \rightarrow R$

Since $Q$ is true and $Q \rightarrow R$ is a premise, we can infer:

• $R$ is true.

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Step 4: Derive $S$ from $R \rightarrow S$

Since $R$ is true and $R \rightarrow S$ is a premise, we can infer:

• $S$ is true. (But we already knew $S$ is true from $P \land S$.)

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Step 5: Derive $R$ from $S \rightarrow R$

Since $S$ is true and $S \rightarrow R$ is a premise, we can infer:

• $R$ is true. (This is consistent with our earlier conclusion that $R$ is true.)

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Step 6: Look for inconsistency

At this point, we have derived:

• $P$, $Q$, $R$, and $S$ are all true.

However, there is no immediate contradiction from these facts alone. The inconsistency arises from the fact that $P \land S$ implies both $P$ and $S$ are true, but the chain of implications $P \rightarrow Q$, $Q \rightarrow R$, and $R \rightarrow S$ leads to the scenario where all statements $P$, $Q$, $R$, and $S$ must be true.

Yet, the circularity between $R$ and $S$ (where $R \rightarrow S$ and $S \rightarrow R$) suggests that $R$ and $S$ cannot independently hold, creating a loop. This loop creates an inconsistency because, under the assumption $P \land S$, the premises cannot all hold without contradiction.

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Thus, the premises are inconsistent.

Would you like further clarification or details?

Related questions:

1. What role does circular reasoning play in detecting inconsistencies in logic?
2. Can you apply this reasoning to a more complex set of premises?
3. How does this logic proof relate to propositional calculus?
4. What are other ways to demonstrate logical inconsistency using formulas?
5. Can a truth table help in demonstrating the inconsistency of these premises?

Tip:

In logical proofs, look for cyclic dependencies between statements as a potential source of inconsistency.