Math Problem Statement

Put the argument in symbols, then use a truth table to determine its validity. If the elephant is angry, the chicken is frightened. The chicken is frightened and the elephant smirks. So, the elephant is angry.

Solution

The problem asks you to express the argument in symbolic logic, then determine its validity using a truth table. The argument is:

Premises:

  1. If the elephant is angry, then the chicken is frightened. (This is p→rp \rightarrow r)
  2. The chicken is frightened and the elephant smirks. (This is r∧qr \land q)

Conclusion: Therefore, the elephant is angry. (This is ∴p\therefore p)

The symbolic form of the argument is:

(p \rightarrow r) \\ r \land q \\ \therefore p \end{aligned}$$ ### Step 1: Create a Truth Table Let's create a truth table for $$p$$, $$q$$, and $$r$$. We need to evaluate the truth of the premises and conclusion under all possible truth values for these three variables. | $$p$$ | $$q$$ | $$r$$ | $$p \rightarrow r$$ | $$r \land q$$ | Conclusion $$p$$ | |------|------|------|-------------------|--------------|--------------| | T | T | T | T | T | T | | T | T | F | F | F | T | | T | F | T | T | F | T | | T | F | F | F | F | T | | F | T | T | T | T | F | | F | T | F | T | F | F | | F | F | T | T | F | F | | F | F | F | T | F | F | ### Step 2: Analyze the Validity An argument is valid if every time the premises are true, the conclusion is also true. In this case: - $$p \rightarrow r$$ is true when $$p$$ is false or both are true. - $$r \land q$$ is true only when both $$r$$ and $$q$$ are true. Looking at the rows where both premises ( $$p \rightarrow r$$ and $$r \land q$$) are true, we see: - Row 1: Both premises are true, and the conclusion $$p$$ is true. - Row 5: Both premises are true, but the conclusion $$p$$ is false. Thus, in row 5, the premises are true, but the conclusion is false. This means the argument is **invalid**. Would you like any further clarification or more details on this process? Here are 5 related questions to think about: 1. What happens when you change the conclusion to something other than $$p$$? 2. Can you construct a counterexample using the truth table? 3. How do logical operators such as $$\land$$ and $$\rightarrow$$ affect the truth values of compound statements? 4. Can an argument with true premises and a false conclusion ever be valid? 5. How do logical fallacies relate to invalid arguments? **Tip:** When constructing truth tables, ensure that all possible combinations of truth values for the variables are represented to avoid missing any critical cases.

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Math Problem Analysis

Mathematical Concepts

Symbolic Logic
Truth Tables
Logical Connectives

Formulas

Implication (p → r)
Conjunction (r ∧ q)

Theorems

Truth Table Construction
Argument Validity

Suitable Grade Level

Undergraduate Logic or Advanced High School

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