Show that the premises W ∧ X, W → Y , X → Z, Y →∼ Z are inconsistent

To show that the premises \( W \land X \), \( W \rightarrow Y \), \( X \rightarrow Z \), and \( Y \rightarrow \sim Z \) are inconsistent, we need to demonstrate that these premises lead to a contradiction. Let's analyze this step by step.

### Premises:
1. \( W \land X \)  (Premise 1: both \( W \) and \( X \) are true)
2. \( W \rightarrow Y \)  (Premise 2: if \( W \) is true, then \( Y \) is true)
3. \( X \rightarrow Z \)  (Premise 3: if \( X \) is true, then \( Z \) is true)
4. \( Y \rightarrow \sim Z \)  (Premise 4: if \( Y \) is true, then \( Z \) is false)

### Step-by-Step Derivation:

1. From premise 1 (\( W \land X \)), we know:
   - \( W \) is true.
   - \( X \) is true.

2. From premise 2 (\( W \rightarrow Y \)) and knowing that \( W \) is true, we can conclude:
   - \( Y \) is true.

3. From premise 3 (\( X \rightarrow Z \)) and knowing that \( X \) is true, we can conclude:
   - \( Z \) is true.

4. From premise 4 (\( Y \rightarrow \sim Z \)) and knowing that \( Y \) is true, we conclude:
   - \( Z \) is false.

### Contradiction:
- In step 3, we derived that \( Z \) is true.
- In step 4, we derived that \( Z \) is false.

This is a contradiction because \( Z \) cannot be both true and false simultaneously. Therefore, the premises are **inconsistent**.

### Summary:
The premises \( W \land X \), \( W \rightarrow Y \), \( X \rightarrow Z \), and \( Y \rightarrow \sim Z \) are inconsistent because they lead to a logical contradiction.

Would you like to go into more detail, or do you have any questions?

Here are 5 related questions:
1. How can consistency in a set of premises be formally defined in logic?
2. What are some methods to check for consistency or inconsistency in logical statements?
3. How does the principle of explosion relate to inconsistent premises?
4. Can an inconsistent set of premises ever be used in a valid logical argument?
5. How would introducing a tautology affect the consistency of the premises?

**Tip:** Always check the final derivation step in logic problems, as this is often where inconsistencies or contradictions become evident.

Discover how to demonstrate inconsistency in logical premises W ∧ X, W → Y, X → Z, and Y → ¬Z with a step-by-step logical analysis. This problem involves logical propositions and contradictions, suitable for advanced learners.

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