Aim: Invalid Arguments Course: Math Literacy Do Now: Aim: What’s an Invalid Argument? Construct a truth table to show the following argument is not valid. If a person reads the Times, then she is well informed. This person is well informed. Therefore, this person reads the Times. An argument is called a valid argument if and only if its premises are true, thereby forcing the conclusion to be true.

Aim: Invalid Arguments Course: Math Literacy Do Now - 1st Invalid Argument pq T T F F T F T F p  qp  q T F T T T F T F T T F T (p  q) ˄ q [(p  q) ˄ q]  p Let p represent “A person reads the Times.” let q represent “This person is well informed.” p  q q  p Not true for all situations – invalid argument This person could be well informed w/o reading the Times. counterexample fallacy of converse or assuming the consequent

Aim: Invalid Arguments Course: Math Literacy Model Problem – 1 st Valid Argument Premises: If I oversleep, then I’ll be late for school. I am late for school. Conclusion: I overslept. Why is this an invalid argument? I could be late for other reasons. Let p: I oversleepq: I am late pq T T F F T F T F p  qp  q T F T T T F T F T T F T (p  q) ˄ q [(p  q) ˄ q]  p

Aim: Invalid Arguments Course: Math Literacy 1st Invalid Argument p  q q  p fallacy of converse or assuming the consequent Both premises ‘p  q’ and q must be true for conclusion to be true because of the conjunction T or F T What is truth of q? F  T is true p could be T or F T T or F T T  T is true p  q:

Aim: Invalid Arguments Course: Math Literacy Model Problem – 1 st Invalid Argument In an episode of the TV series “Star Trek”, the starship Enterprise is hit by an ion storm, causing the power to go out. Captain Kirk wonders if Mr. Scott, the engineer, is aware of the problem. Mr. Spock, the paragon of extraterrestrial intelligence, replies, “If Mr. Scott is still with us, the power should be on momentarily.” Moments later, the ship’s power comes on and Spock arches his Vulcan brow: “Ah, Mr. Scott is still with us.” Express Spock’s statements as an argument. If Mr. Scott is still with us, the power will come on. The power comes on. Therefore, Mr. Scott is still with us. valid or invalid?

Aim: Invalid Arguments Course: Math Literacy Model Problem – 1 st Invalid Argument If Mr. Scott is still with us, the power will come on. The power comes on. Therefore, Mr. Scott is still with us. p: Mr. Scott is still with us. q: Power will come on. 1.Use a letter to represent each simple statement in the argument. 2. Express the premises and the conclusion symbolically. p  q q  p Someone other than Scottie could have started the engines fallacy of converse or assuming the consequent

Aim: Invalid Arguments Course: Math Literacy 2 nd Invalid Argument p  q ~p  ~q This person could be well informed w/o reading the Times. If a person reads the Times, then he is well informed. This person does not read the Times. Therefore, this person is not well informed. fallacy of inverse or denying the antecedent Let p represent “A person reads the Times.” let q represent “This person is well informed.” If ‘p  q’ and ‘~p’, then ‘~q’. [(p  q) ˄ ~p]  ~q

Aim: Invalid Arguments Course: Math Literacy 2 nd Invalid Argument If a person reads the Times, then he is well informed. This person does not read the Times. Therefore, this person is not well informed. Let p represent “A person reads the Times.” let q represent “This person is well informed.” [(p  q) ˄ ~p]  ~q pq~p~q T T F F T F T F p  qp  q T F T T F F T T T T F T (p  q) ˄ ~p [(p  q) ˄ ~p]  ~q F F T T F T F T

Aim: Invalid Arguments Course: Math Literacy 2 nd Invalid Argument p  q ~p  ~q fallacy of inverse or denying the antecedent Both premises ‘p  q’ and ~p must be true for conclusion to be true because of the conjunction F ~F What is truth of q? F  T is true F  F is true ? q could be T or F ~q could be T or F T or F

Aim: Invalid Arguments Course: Math Literacy 2 nd Invalid Argument Example p  q: If x = 10, then x is greater than 5. ~p: x  10  ~q: x is not greater than 5 If x = 3, then it is T that ‘3 is not greater than 5’..... but If x = 8, then it is F that ‘8 is not greater than 5’.... consequently We cannot conclude that ~q, which means ‘x is not greater than 5’, is true. pq~p~q T T F F T F T F p  qp  q T F T T F F T T T T F T (p  q) ˄ ~p [(p  q) ˄ ~p]  ~q F F T T F T F T

Aim: Invalid Arguments Course: Math Literacy Model Problem – 2 nd Valid Argument If a person goes to college, he will make a lot of money. You don’t go to college. Therefore, you will not make a lot of money. Test the validity of the following Let p: person goes to college q: he will make a lot of money p  q ~p ~q p  q ~p  ~q fallacy of inverse or denying the antecedent You could win the lottery.

Aim: Invalid Arguments Course: Math Literacy Model Problem – 2 nd Invalid Argument If I study, I pass. I do not study. Therefore, I do not pass. Test the validity of the following Let p: I study q: I pass p  q ~p ~q~q p  q ~p  ~q Some people are just super smart! fallacy of inverse or denying the antecedent

Aim: Invalid Arguments Course: Math Literacy Reasoning and Fallacies If the argument is valid. state the law of reasoning that tells why the conclusion is true. If the argument is invalid, say so. 2 nd invalid fallacy of inverse Law of Detachment 1 st invalid fallacy of converse Law of Modus Tollens Law of Contrapositive

Aim: Invalid Arguments Course: Math Literacy Testing Validity 1.Use a letter to represent each simples statement in the argument. 2.Express the premises and the conclusion symbolically. 3.Write a symbolic conditional statement is the form [(premise 1) ˄ (premise 2) ˄...]  Conclusion 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 not all true, argument is not valid.

Aim: Invalid Arguments Course: Math Literacy Model Problem – 2 nd Invalid Argument