Section 3 5 Arguments and Truth Tables Objectives

Section 3. 5 Arguments and Truth Tables Objectives 1. Use truth tables to determine validity. 2. Recognize and use forms of valid and invalid arguments. 1/24/2022 Section 3. 6 1

Arguments • An Argument consists of two parts: – Premises: the given statements. – Conclusion: the result determined by the truth of the premises. • Valid Argument: The conclusion is true whenever the premises are assumed to be true. • Invalid Argument: Also called a fallacy • Truth tables can be used to test validity. 1/24/2022 Section 3. 6 2

Example 1 The Menendez case Did Eric and Lyle Menendez kill their parents to get the inheritance or as a result of years of abuse? Premise 1: If children murder their parents in cold blood, they deserve to be punished to the full extent of the law. . Premise 2: These children murdered their parents in cold blood. Conclusion: Therefore, these children deserve to be punished to the full extent of the law. 1/24/2022 Section 3. 6 3

Example 1 continued The Menendez case • Solution p: Children murder their parents in cold blood. q: They deserve to be punished to the full extent of the law. Write the argument in symbolic form: Premise 1: p q If children under their parents in cold blood, they deserve to be punished to the full extent of the law. Premise 2: p These children murdered their parents in cold blood. Conclusion: q Therefore, these children deserve to be punished to the full extent of the law. 1/24/2022 Section 3. 6 4

Example 1 continued Rewriting as a conditional statement and forming the truth table: [(p q) p] q This conditional statement is a tautology. This argument is called direct reasoning and is valid. 1/24/2022 p q (p q) p [(p q) p] q T T T F F F T T F F T Section 3. 6 5

Testing the validity of an Argument with a Truth Table 1. Use a letter to represent each simple statement in the argument. 2. Express the premises and the conclusion symbolically. 3. Write a symbolic conditional statement of the form: [(premise 1) (premise 2) … (premise n)] conclusion, where n is the number of premeses. 4. Construct a truth table for the conditional statement in step 3. 5. If the final column of the truth table has all trues, the conditional statement is a tautology and the argument is valid. If the final column does not have all trues, the conditional statement is not a tautology and the argument is invalid. 1/24/2022 Section 3. 6 6

Example 2 Determining Validity with a Truth Table • Determine if the following argument is valid: “I can’t have anything more to do with the operation. If I did, I’d have to lie to the Ambassador. And I can’t do that” —Henry Bromell • Solution: We can express the argument as follows: If I had anything more to do with the operation. I’d have to lie to the Ambassador. I can’t lie to the Ambassador. Therefore, I can’t have anything more to do with the operation. 1/24/2022 Section 3. 6 7

Example 2 continued Step 1: Use a letter to represent each statement in the argument: p: I have more to do with the operation q: I have to lie to the Ambassador. Step 2: Express the premises and the conclusion symbolically. p q If I had anything more to do with the operation, I’d have to lie to the Ambassador. ~q I can’t lie to the Ambassador. . ~p Therefore, I can’t have anything more to do with the operation. 1/24/2022 Section 3. 6 8

Example 2 continued Step 3. Write a symbolic statement: [(p q) ~q ] ~ p Step 4. Construct the truth table. The form of this argument is called contrapositive reasoning. It is a valid argument. p q p q ~q (p q) ~q ~ p [(p q) ~q ] ~ p T T T F F F T T F F T T T 1/24/2022 Section 3. 6 9

Standard Forms of Arguments Valid Arguments Direct Contrapositive Reasoning p q p____ ~q____ q ~p Invalid Arguments Fallacy of the Converse Inverse p q q ~p p ~q 1/24/2022 Disjunctive Reasoning p q ~p ~q q p Misuse of Disjunctive Reasoning p q p q ~p Section 3. 6 Transitive Reasoning p q q r p r ~p Misuse of Transitive Reasoning p q q r r p ~p ~r 10

Example 3 Determining Validity Without Truth Tables • Determine whether this argument is valid or invalid: There is no need for surgery because if there is a tumor then there is a need for surgery but there is no tumor. • Solution: p: There is a tumor q: There is a need for surgery. Expressed symbolically: If there is a tumor then there is need for surgery. p q There is no tumor. . ~p Therefore, there is no need for surgery. ~q The argument is in the form of the fallacy of the inverse and is therefore, invalid. 1/24/2022 Section 3. 6 11