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Use the TKP method to check the validity of the following arguments 

To check the validity of the given argument using the **TKP (Tableau or Truth Tree) Method**, we will proceed step-by-step:

### Problem:
You are given a set of premises and a conclusion in the following format:
1. \( P_1: \neg [(a \lor b) \to \neg c] \)
2. \( P_2: \neg d \lor \neg e \lor \neg c \)
3. \( P_3: a \to d \)
4. \( P_4: e \)
5. Conclusion (K): \( b \)

The goal is to determine whether the argument is valid using the **Tableau Method** by assuming the premises are true and the conclusion is false, and then attempting to derive a contradiction.

### Steps:

1. **Negation of the conclusion:**
   We negate the conclusion, \( K: b \), to begin the tableau:
   \[
   \neg b
   \]
   
2. **Starting from the premises:**
   We consider the premises as given:
   - \( P_1: \neg [(a \lor b) \to \neg c] \)
   - \( P_2: \neg d \lor \neg e \lor \neg c \)
   - \( P_3: a \to d \)
   - \( P_4: e \)

3. **Breaking down the logical structure:**
   - From \( P_1: \neg [(a \lor b) \to \neg c] \), we use the equivalence of implications to expand it as:
     \[
     \neg [\neg (a \lor b) \lor \neg c]
     \]
     This negation simplifies to:
     \[
     (a \lor b) \land c
     \]
     So, we have:
     \[
     a \lor b
     \quad \text{and} \quad c
     \]
   
   - From \( P_3: a \to d \), we can rewrite this as:
     \[
     \neg a \lor d
     \]
   
   - Premise \( P_4: e \) is simply true.

4. **Analyzing the branches:**
   Now we proceed by constructing the truth tree from the assumptions:
   - We have \( \neg b \), so \( b \) is false.
   - From \( a \lor b \) and \( \neg b \), it follows that \( a \) must be true.
   - If \( a \) is true, then from \( P_3: a \to d \), it follows that \( d \) must also be true.
   - From \( P_2: \neg d \lor \neg e \lor \neg c \), given that \( d \) and \( e \) are true, it must be that \( \neg c \) is true, meaning \( c \) is false.

5. **Contradiction:**
   At this point, we have both \( c \) and \( \neg c \) being true, which is a contradiction.

### Conclusion:
Since assuming the premises to be true and the conclusion false leads to a contradiction, the argument is valid.

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Would you like further details or any clarifications?

Here are five relative questions to expand the understanding:
1. How does the **Tableau method** differ from truth tables in proving validity?
2. Can you apply the **contrapositive** rule to any premise to check the argument's structure?
3. What are some common logical equivalences useful in simplifying logical statements?
4. How does the tableau method handle **disjunctions** or multiple branches in an argument?
5. What other methods (e.g., **Natural Deduction**) can be used to prove the validity of an argument?

**Tip:** When applying the Tableau method, always look for immediate contradictions (like \( c \) and \( \neg c \)) to quickly verify validity.

Use the TKP method to check the validity of the following arguments.

Learn how to apply the TKP (Tableau) method to verify the validity of a logical argument with multiple premises and a conclusion. Explore key concepts such as logical implication, negation, and contradiction in propositional logic.

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