Analyzing Arguments with Truth Tables
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1 Analyzing Arguments with Truth Tables MATH 100 Survey of Mathematical Ideas J. Robert Buchanan Department of Mathematics Fall 2014
2 Introduction Euler diagrams are useful for checking the validity of simple arguments, especially if they involve universal ( all, every, etc.) or existential ( some, etc.) qualifiers. Today we will learn how to use truth tables to test the validity of arguments. Truth tables are generally used on more complex arguments.
3 Example (1 of 2) Consider the logical argument: If Johnny Forbes sells his quota, he gets a bonus. Johnny Forbes sells his quota. He gets a bonus.
4 Example (1 of 2) Consider the logical argument: If Johnny Forbes sells his quota, he gets a bonus. Johnny Forbes sells his quota. He gets a bonus. Identify the component statements in the argument. p: Johnny Forbes sells his quota. q: He gets a bonus.
5 Example (1 of 2) Consider the logical argument: If Johnny Forbes sells his quota, he gets a bonus. Johnny Forbes sells his quota. He gets a bonus. Identify the component statements in the argument. p: Johnny Forbes sells his quota. q: He gets a bonus. Write the argument in symbolic form. Premise 1: p q Premise 2: p Conclusion: q
6 Example (2 of 2) Premise 1: p q Premise 2: p Conclusion: q We must decide if the conjunction of the premises implies the conclusion. [(p q) p] q We do this with a truth table.
7 Example (2 of 2) Premise 1: p q Premise 2: p Conclusion: q We must decide if the conjunction of the premises implies the conclusion. [(p q) p] q We do this with a truth table. p q p q (p q) p [(p q) p] q T T T T T T F F F T F T T F T F F T F T
8 Modus Ponens The pattern of the argument, p q p q occurs frequently and is called the law of detachment or modus ponens (Latin for the way that affirms by affirming ).
9 General Procedure To test the validity of an argument using a truth table following these steps: Step 1: assign a letter to each component statement of the argument. Step 2: express each premise and the conclusion symbolically. Step 3: form the symbolic statement of the entire argument by writing the conjunction of all the premises as the antecedent and the conclusion of the argument as the consequent. Step 4: complete the truth table for the conditional statement. If it is a tautology, the argument is valid; otherwise, it is invalid.
10 Example (1 of 2) Determine whether the following argument is valid or invalid. If I could be in two places at one time, I d be at the movies. I am not at the movies. I can t be in two places at one time.
11 Example (1 of 2) Determine whether the following argument is valid or invalid. If I could be in two places at one time, I d be at the movies. I am not at the movies. I can t be in two places at one time. Component statements: Symbolic form: p: I can be in two places at one time. q: I am at the movies. Premise 1: p q Premise 2: q Conclusion: p
12 Example (2 of 2) Premise 1: p q Premise 2: q Conclusion: p Truth table: p q p q p q (p q) q [(p q) q] p T T T F F F T T F F F T F T F T T T F F T F F T T T T T
13 Example (2 of 2) Premise 1: p q Premise 2: q Conclusion: p Truth table: p q p q p q (p q) q [(p q) q] p T T T F F F T T F F F T F T F T T T F F T F F T T T T T The argument is valid.
14 Example Use a truth table to determine the validity of the following argument. If Nelson Dida plays, the opponent gets shut out. The opponent does not get shut out. Nelson Dida does not play. Use your i>clicker2 to indicate whether the argument is A: Valid B: Invalid
15 Valid Argument Forms We will often see the following patterns in valid arguments. Modus Modus Disjunctive Reasoning by Ponens Tollens Syllogism Transitivity p q p q p q q p p q p q p q q r p r
16 Valid Argument Forms We will often see the following patterns in valid arguments. Modus Modus Disjunctive Reasoning by Ponens Tollens Syllogism Transitivity p q p q p q q p p q p q p q q r p r Modus tollens is Latin for the way that denies by denying.
17 Example Use a truth table to determine the validity of the following argument. Mia kicks or Arnold pumps iron. Arnold does not pump iron. Mia kicks. Select the appropriate response from the following list using your i>clicker2. A: Valid by modus ponens. B: Valid by modus tollens. C: Valid by disjunctive syllogism. D: Valid by reasoning by transitivity. E: Invalid
18 Invalid Argument Forms (Fallacies) There are also commonly seen invalid forms of argument. Fallacy of the Converse p q q p Fallacy of the Inverse p q p q
19 Explore the Fallacy of the Converse p q q p p q p q (p q) q ((p q) q) p T T T T T T F F F T F T T T F F F T F T This is not a tautology.
20 Example Use a truth table to determine the validity of the following argument. If we buy groceries, then we must be hungry. We do not buy groceries. We are not hungry. Select the appropriate response from the following list using your i>clicker2. A: Valid by modus ponens. B: Valid by modus tollens. C: Valid by disjunctive syllogism. D: Invalid by the fallacy of the converse. E: Invalid by the fallacy of the inverse.
21 More Than Two Premises Consider the argument: A (p q) (q r) B p r C r p We check its validity with a larger truth table (some columns omitted). p q r A B A B C (A B) C T T T F T F T T T T F T T T T T T F T T T T T T T F F T T T T T F T T T F F F T F T F T T T T T F F T T F F F T F F F T T T T T
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