3.7 Arguments and Truth Tables

1. 3.7 Arguments and Truth Tables

2. An argument consist of two parts: the given statements, called the premises, and a conclusion. An argument is valid if the conclusion is true whenever the premises are assumed to be true. An argument that is not valid is said to be an invalid argument, also called a fallacy. If a truth table shows a tautology, then the argument is valid.

3. A noted criminal case in 1995 involved Lyle and Erik Menendez, who shot and killed their parents. Although everyone agreed that the brothers committed the crime, it took two trials before they were convicted. The arguments in the trial centered around the boys’ motivation: Was the killing a premeditated act by two children hoping to receive an inheritance, or was it an act motivated by years of abuse and a desperate sense of helplessness and rage? Here is the prosecutor’s argument from the Menendez brothers’ criminal case

4. p: Children murder their parents in cold blood. q: They deserve to be punished to the full extent of the law. 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. [(p → q) ∧ p] → q

5. [(p → q) ∧ p] → q p q p → q (p → q) ∧ p [(p → q) ∧ p] → q T T T T T T F F F T F T F T T T F T F F The argument is valid. This is called direct reasoning.

6. Use a letter to represent each simple statement in the argument. Express the premises and the conclusion symbolically. Write a symbolic conditional statement of the form [(premise 1) ∧ (premise 2) ∧ … ∧ (premise n)] → conclusion Construct a truth table for the conditional statement in step 3. If the final column is all trues, then the argument is valid. Otherwise it is invalid.

8. Premise 1: If Mr. Scott is still with us, then the power will come back on. Premise 2: The power comes on. Conclusion: Mr. Scott is still with us. p: Mr. Scott is still with us. q: The power will come back on. [(p → q) ∧ q] → p

9. [(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 F T T F T F F The argument is invalid. (Spock blew it.) This is called the fallacy of the converse.

10. If your heart is beating, then you are alive. You are not alive. Your heart is not beating. Contrapositive reasoning Let p represent “Your heart is beating.” Let q represent “You are alive.” ∼q ∼p [(p→q)∧∼q]→∼p p→ q (p→ q) ∧∼q p q F T F F T T T T F F F T T F F T F T T F T T T T T T F F

11. If your heart is beating, then you are alive. You heart is not beating. You are not alive. Fallacy of the inverse Let prepresent “Your heart is beating.” Let q represent “You are alive.” ∼p ∼q [(p→q)∧∼p]→∼q p→ q (p→ q) ∧∼p p q F T F F T T T F F F T T T F T T T F F F T T T T T T F F

12. If you are human, then you walk on two legs. If you walk on two legs, then you can play the piano. If you are human, then you can play the piano. Let p represent “You are human.” Let q represent “You walk on two legs.” Let r represent “You can play the piano.” Transitive Reasoning [(pq)(qr)](pr) p q pq q r (pq)(qr) pr r T T T T T T T T T F F F T T T F T T T F F F T T F F T F F T T F T T T T T T F T F T T T F F T F T T T T T T F F T T T T T F F F

13. Valid Arguments p→ q p ∴ q p→ q ∼q ∴ ∼p p∨ q ∼p ∴ q p∨ q ∼q ∴ p Disjunctive Direct Contrapositive p→ q q→ r ∴ p→ r ∴ ∼r→ ∼p Transitive

14. Invalid Arguments p→ q q ∴ p p→ q ∼p ∴ ∼q p∨ q q ∴ ∼p p∨ q p ∴ ∼q Misuse of Disjunctive Fallacy of the Converse Fallacy of the Inverse p→ q q→ r ∴ r→ p ∴ ∼p→ ∼r Misuse of Transitive

15. p→ q p ∴ q p→ q ∼q ∴ ∼p p∨ q ∼p ∴ q p∨ q ∼q ∴ p p→ q q→ r ∴ p→ r ∴ ∼r→ ∼p Contrapositive Direct Disjunctive Transitive The emergence of democracy is a cause for hope or environmental problems will overshadow any promise of a bright future. Environmental problems will not overshadow any promise of a bright future. Therefore, the emergence of democracy is a cause for hope. d: The emergence of democracy is a cause for hope. e: Environmental problems will overshadow any promise of a bright future. d∨ e ∼e ∴ d

16. p→ q p ∴ q p→ q ∼q ∴ ∼p p∨ q ∼p ∴ q p∨ q ∼q ∴ p p→ q q→ r ∴ p→ r ∴ ∼r→ ∼p Contrapositive Direct Disjunctive Transitive If the defendant’s DNA is found at the crime scene, then we can have him stand trial. He is standing trial Consequently, we found evidence of his DNA at the crime scene. d: The defendants DNA is found at the crime scene. t: We can have him stand trial. Fallacy of the Converse

17. p→ q p ∴ q p→ q ∼q ∴ ∼p p∨ q ∼p ∴ q p∨ q ∼q ∴ p p→ q q→ r ∴ p→ r ∴ ∼r→ ∼p Contrapositive Direct Disjunctive Transitive If you mess up, your self-esteem goes down. If your self-esteem goes down, everything else falls apart. So, if you mess up, everything else falls apart. m: You mess up. s: Your self-esteem goes down. e: Everything else falls apart.

18. If , then If , then If , then

19. If , then If , then If , then

20. If , then If , then If , then

21. If , then If , then If , then If , then Therefore

22. What happened? Our logic was flawless. The premises must be true for the conclusion to be true. The argument is valid. The conclusion is not true because our first premise was false. If , then If , then