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:

     \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.

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.