6. Prove that the premises 𝑃 → 𝑄, 𝑄 → 𝑅, 𝑅 → 𝑆, 𝑆 → 𝑅 𝑎𝑛𝑑 𝑃 ∧ 𝑆 are

inconsistent. only using formulas

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:

---

### Step 1: Start with \( P \land S \)
From the premise \( P \land S \), we know:
- \( P \) is true.
- \( S \) is true.

---

### 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.

---

### 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.

---

### 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 \( P \land S \).)

---

### 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.)

---

### 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.

---

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.

Prove that the premises P → Q, Q → R, R → S, S → R, and P ∧ S are inconsistent, using formulas.

Explore how to prove the inconsistency of logical premises: P → Q, Q → R, R → S, S → R, and P ∧ S. This problem focuses on propositional logic and identifies logical contradictions using formulas. Ideal for college-level logic and discrete mathematics students.

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