https://file.aiquickdraw.com/mathbot/file/1727197819776ad6lisc8.jpg

The argument in the image involves symbolic logic and truth tables to assess validity. Here's how to approach it:

### Given Statements:
1. **P** = "Elephant is angry."
2. **Q** = "Elephant smirks."
3. **R** = "Chicken is frightened."

### Argument in symbols:
- **Premise 1**: If the elephant is angry, the chicken is frightened.  
  This can be written as: \( P \rightarrow R \)
  
- **Premise 2**: The chicken is frightened, and the elephant smirks.  
  This can be written as: \( R \land Q \)
  
- **Conclusion**: So, the elephant is angry.  
  This can be written as: \( P \)

### Full Argument Structure:
1. \( P \rightarrow R \) (Premise)
2. \( R \land Q \) (Premise)
3. \( P \) (Conclusion)

### Step-by-Step Truth Table:

We need to construct a truth table that evaluates whether the conclusion (\( P \)) logically follows from the premises.

| **P** | **Q** | **R** | \( P \rightarrow R \) | \( R \land Q \) | Conclusion \( P \) |
|-------|-------|-------|---------------------|-----------------|-----------------|
| T     | T     | T     | T                   | T               | T               |
| T     | T     | F     | F                   | F               | T               |
| T     | F     | T     | T                   | F               | T               |
| T     | F     | F     | F                   | F               | T               |
| F     | T     | T     | T                   | T               | F               |
| F     | T     | F     | T                   | F               | F               |
| F     | F     | T     | T                   | F               | F               |
| F     | F     | F     | T                   | F               | F               |

### Conclusion:
The argument is **not valid** because there are rows where both premises are true (e.g., the last row where \( R \land Q \) and \( P \rightarrow R \) are true) but the conclusion \( P \) is false.

Would you like more details or have any questions?

### Related Questions:
1. How do logical implications work in symbolic logic?
2. How is the truth table method used to test an argument’s validity?
3. What are common forms of logical fallacies in deductive reasoning?
4. How can premises with conjunctions (\( \land \)) affect the conclusion?
5. How does symbolic logic apply to real-life arguments?

**Tip**: When evaluating logical arguments, always ensure the conclusion must hold in all possible truth scenarios to validate the argument.

If the elephant is angry, the chicken is frightened. The chicken is frightened and the elephant smirks. So, the elephant is angry.

This problem explores the validity of a logical argument using symbolic logic and truth tables. Learn how to analyze an argument involving elephants and chickens, and how truth tables help determine the validity of logical conclusions.

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