aim: invalid arguments course: math literacy do now: aim: what’s an invalid argument? construct a...

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.


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


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


1st Invalid Argument

p q

q

p



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:


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?


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




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


If ‘p q’ and ‘~p’, then ‘~q’.

[(p q) ˄ ~p] ~q



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.


[(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



p q

~p

~q



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


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


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



You could win the lottery.


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!




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


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.


Model Problem – 2nd Invalid Argument

aim: invalid arguments course: math literacy do now: aim: what’s an invalid argument? construct a...

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