Let's work through the problem again, ensuring clarity in notation and correct steps while applying the Tableau method (TKP method) with your given restated premises.

Premises and Goal:

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. $P_4: e$

Goal: Prove $b$

Negating the Conclusion:

To use the Tableau method, we negate the conclusion: $\neg K: \neg b$

Step-by-Step Analysis Using the Tableau Method:

We aim to show that the premises $P_1, P_2, P_3, P_4$ combined with the negation of the conclusion $\neg b$ lead to a contradiction.

If this happens, we can conclude the argument is valid.

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Step 1: Analyze $P_1$

Premise $P_1: \neg [(a \lor b) \to \neg c]$.

• First, we simplify $(a \lor b) \to \neg c$. Using the implication equivalence $p \to q \equiv \neg p \lor q$, we rewrite: $(a \lor b) \to \neg c \equiv \neg (a \lor b) \lor \neg c$
• Now, apply the negation from $P_1$: $\neg [\neg (a \lor b) \lor \neg c] \equiv (a \lor b) \land c$ Thus, $P_1$ gives us:
• $a \lor b$ is true.
• $c$ is true.

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Step 2: Analyze $P_2$

Premise $P_2: \neg d \lor \neg e \lor \neg c$.

From $P_1$, we know that $c$ is true, so $\neg c$ is false. Therefore, we can simplify $P_2$ to: [ P_2: \neg d \