1. Analyzing Arguments Section 1.5

2. Valid arguments An argument consists of two parts: the hypotheses (premises) and the conclusion. An argument is valid if the conclusion of the arguments is guaranteed under the given set of hypotheses.

3. Conditional Representation of an Argument An argument having n hypotheses, h 1, h 2, …, h n and conclusion c can be represented by the conditional [h 1 ^ h 2 ^ … ^ h n ]  c. If the above conditional is always true, (regardless of the truthfulness of the individual statements) the argument is valid.

4. Tautologies A tautology is a statement that is always true. What this means is that if every entry for a particular column in a truth table has a value of true, then that statement is a tautology. An argument having n hypotheses h 1, h 2, …, h n and conclusion c is valid if and only if the conditional [h 1 ^ h 2 ^ … ^ h n ]  c is a tautology.

5. Example Is this argument valid? If you listen to rock and roll, you do not go to heaven. If you are a moral person, you go to heaven. Therefore, you are not a moral person if you listen to rock and roll. Step 1: Identify the hypotheses and the conclusion. Step 2: Identify the simple statements in the hypotheses and conclusion. Step 3: Write the hypotheses and conclusion in symbolic form. Step 4: Construct a truth table. Step 5: Verify if the conditional [h 1 ^ h 2 ^ … ^ h n ]  c is a tautology.

6. Example – Step 1 Step 1:Identify the hypotheses and conclusion. h 1 : If you listen to rock and roll, you do not go to heaven. h 2 : If you are a moral person, you go to heaven. c: Therefore, you are not a moral person if you listen to rock and roll.

7. Example – Step 2 Step 2: Identify the simple statements. p: You listen to rock and roll. q: You go to heaven. r: You are a moral person. Note: 3 simple statements implies 8 rows in the truth table.

8. Example – Step 3 Step 3: Write the hypotheses and conclusion in symbolic form. h 1 : p  ~q h 2 : r  q c : p  ~r (remember the conclusion is using the if connective ~r if p.)

10. Example – Step 5 Step 5: Is h 1 ^ h 2  c a tautology. Yes. Looking at the last column of the truth table, we see that all the values are TRUE. So the argument is valid.