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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
p q
T
T
F
F
T
F
T
F
p 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 – 1st 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 oversleep q: I am late
p q
TTF
F
TFT
F
p q
TFT
T
TFT
F
TTF
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 truep q:
Aim: Invalid Arguments Course: Math Literacy
Model Problem – 1st 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 – 1st 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
q
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
2nd 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
2nd 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
p q ~p ~q
T
T
F
F
T
F
T
F
p 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
2nd 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
T or F
Aim: Invalid Arguments Course: Math Literacy
2nd 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.
p q ~p ~q
T
T
F
F
T
F
T
F
p 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 – 2nd 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 – 2nd 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
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.
~
~
p q
p
q
2nd invalid fallacy of
inverse
p q
p
q
Law of Detachment
p q
q
p
1st invalidfallacy of converse
~
~
p q
q
p
Law of Modus Tollens
~ ~
p q
q p
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 – 2nd Invalid Argument