c]*P*1:¬[(*a*∨*b*)→¬*c*]

P2:¬d∨¬e∨¬c*P*2:¬*d*∨¬*e*∨¬*c*

P3:a→d*P*3:*a*→*d*

P4:e

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.

---

### 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.

---

### 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 \

P1: ¬[(a ∨ b) → ¬c] P2: ¬d ∨ ¬e ∨ ¬c P3: a → d P4: e

Explore how to use the Tableau method to prove the validity of an argument in propositional logic. This problem involves premises like P1: ¬[(a ∨ b) → ¬c] and applies key concepts such as implication equivalence and contradiction. Ideal for students in undergraduate logic courses.

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