Determining Validity and Counterexamples

Determining Validity and Counterexamples

by Mrs. C Moore

​Is it possible to draw the wrong conclusion even when you are given true information?

​Look at this example:

​A man walks into a barbershop where there is a dog chained up nearby. He asks the barber "Does your dog bite?" The barber replies, " Oh Sammy, he wouldn't hurt a fly." So the man reaches over to pat the dog and the dog almost takes off his hand! The emergency response team is called and the man yells to the barber from the stretcher, "I thought you said your dog didn't bite!" The barber replied, "That's not my dog."

​On the following slides, we will look at more counterexamples as well as logic laws as we seek to test the validity of logical arguments.

​A Counterexample is a statement that proves that the conclusion to a conditional statement is false.

​False conditional statement:

​

​If the dog is chained in the barber shop, then it must belong to the barber.

​

Whose dog could it have been?

​

​If that was the barber's dog, then it would not have bitten the man.

​

​It did bite the man

​

​Therefore, it was not the barber's dog.

​

​This is a valid argument. Which law does this argument follow?

Multiple Choice

Which of the following is a counterexample of the statement:

"If an animal is a bird, then it can fly."

• seagull

• penguin

• robin

• eagle

Multiple Select

Which of the following are counterexamples of the statement:

"If a number is an odd number, then it is a prime number."

• 31

• 24

• 2

• 9

Multiple Choice

Which is a counterexample to the following statement? If an angle is obtuse, then it is $$125\degree$$ .

• $$90\degree$$  

• $$57\degree$$  

• $$160\degree$$  

• $$180\degree$$  

Multiple Choice

Provide a counterexample to the following claim:

"All quadrilaterals have at least one set of parallel sides."

• Trapezoid

• Rectangle

• Rhombus

• $$180\degree$$  

Multiple Choice

If it is a fraction, then it is a number between zero and one. What is an appropriate counterexample?

• 2/3
• 4/5
• 1/2
• 3/2

Multiple Choice

Which is a counterexample to the following statement?

If an animal is brown, then it is a dog.

• Squirrel

• Chocolate lab

• White rabbit

• Blue jay

Multiple Choice

Which is a counterexample to the following statement?

If you add two numbers, then the sum is odd.

• 2+5=7

• 3+2=5

• 1+8=9

• 2+4=6

Multiple Choice

Which is a counterexample of: If a four sided shape has two sides the same length, then it must be a rectangle.

• quadrilateral
• rectangle
• square
• triangle

Multiple Choice

Which answer would be a counterexample to the following biconditional?

"Greg is a swimmer if and only if he is an athlete."

• Greg is an athlete and a basketball player.

• Greg is a swimmer.

• Greg is not an athlete

• There is no counterexample.

Multiple Select

Which is a counterexample to the following statement?

If you live in Texas, then you live in Carrollton.

• New York City

• Dallas

• Houston

• Los Angeles

Multiple Choice

If an animal is furry, then it is a hamster. What would be an appropriate counterexample?

• goldfish
• frog
• elephant
• cat

Multiple Choice

If it is an angle, then it is acute. What is an appropriate counterexample?

• 70⁰
• 45⁰
• 120⁰
• True. There is no counterexample.

Multiple Choice

Which of these is a counterexample to the following statement?

If a number is divisible by 3 then it is odd.

• 3

• 6

• 8

• 11

Multiple Choice

Which of these is a counterexample to the following statement?

Months in the Spring start with a consonant

• December

• March

• April

• October

Multiple Choice

Which of these is a counterexample to the following statement?

All farm animals have four legs.

• Cows

• Penguins

• Elephants

• Chickens

Multiple Choice

Which of these is a counterexample to the following statement?

All vegetables are green.

• Banana

• Pumpkins

• Green Beans

• Grass

Multiple Choice

Which of these is a counterexample to the following statement?

If something is used for transportation then it has wheels.

• Car

• Motorcycle

• Toy Cars

• Boat

Multiple Select

If you pass this class,

then you know everything!!

The truth value of the biconditional would be... (choose all that could apply)

• TRUE - Geometry is Awesome!

• TRUE - if you know everything, you know Geometry.

• FALSE - What about Algebra 2?

• FALSE - You could forget a random detail, but still pass

Multiple Choice

If a shape is round, then it is a circle. What is an appropriate counterexample?

• hexagon
• oval
• triangle
• square

Multiple Choice

Use the law of detachment to draw a conclusion to the given conditional statement:

If a triangle has three equal sides, then the triangle is an equilateral triangle.

Triangle ABC has three equal sides.

• Therefore, it is not an equilateral triangle.

• Therefore, it is a scalene triangle.

• Therefore, it is an equilateral triangle.

• The statement is not valid.

Multiple Choice

Determine whether the stated conclusion is valid.
Given: If an animal is a dog, then they like biscuits.
Sammy is a dog.
Conclusion: Sammy likes biscuits.

• Invalid
• Valid
• Sammy is a Great Dane
• Sammy is really a cat.

Multiple Choice

Draw a conclusion from the statement.
If a parasite lives in or on a host organism, then it harms its host.
A virus is a parasite.

• It will harm the host.
• I have a cold.
• It lives in or on a host
• Invalid

Multiple Choice

Draw a conclusion from the statement.
If Jack goes to school, then it is Monday.
Jack doesn't go to school.

• Jack skipped school.
• It isn't Monday.
• It's Saturday.
• Invalid

Multiple Choice

Use the Law of Detachment to draw a conclusion from the two given statements. If not possible, write not possible.
Statement 1: If I study for one hour each day, then I will score well on the exam
Statement 2: I will score well on the exam

•  I study for 1 hour each day.
• I do not study for 1 hour each day
• If I do well on the exam, I studied for 1 hour each day.
• not possible

Multiple Choice

Using the Law of Syllogism, which of the following completes the statement to form a valid conclusion?
If it is snowing heavily, then school will be canceled.
If school is canceled, the big test will not be given today.
It is snowing heavily, therefore

• look out for the snowplows while driving to school.
• the big test will not be given today.
• the roads will be hard to drive on.
• you should call the school to see if school is canceled.

Multiple Choice

Which law of logic is this:
If I buy a dress, then I won't get chips.
I bought a dress.
Therefore, I won't get chips.

• Law of Detachment
• Law of Contrapositive
• Law of Syllogism
• Law of Biconditional

Multiple Choice

p→q
q→r
∴p→r
Which law is modeled?

• Law of Syllogism
• Law of Detachment
• Law of Contrapositive
• Law of Shannon

Multiple Choice

p→q
p
∴q
Which law is modeled?

• Law of Syllogism
• Law of Detachment
• Law of Contrapositive
• Law of Shannon

Multiple Choice

Draw a conclusion from the statement.
If you are at least 25 year old, then you can rent a car.
You can rent a car

• You are at least 25 years old
• You can rent a car
• You can go on a road trip
• Invalid

Multiple Choice

Let a represent "It is hot outside".
Let b represent "I will eat ice cream".
What is the symbolic notation of "If I don't eat ice cream, then it is not hot outside."

• b -> a
• ~b -> ~a
• ~a -> ~b
• a -> b

Multiple Choice

(1) If you drive safely, then the life you save may be your own.
(2) AJ drives safely.
(3)  The life AJ saves may be his own.
Is statement (3) logical given the Law of Detachment, Syllogism, Contrapositive, or is it invalid?

• Law of Detachment
• Law of Syllogism
• Law of Contrapositive
• invalid

Multiple Choice

Refer to the following statement: "If a polygon is a quadrilateral, then it is a trapezoid."

What is a counterexample of this statement?

• If a polygon is a trapezoid, then it is a quadrilateral.

• If a polygon is not a quadrilateral, then it is not a trapezoid.

• If a polygon is not a trapezoid, then it is not a quadrilateral.

• A rectangle is also a quadrilateral.

Multiple Choice

Using the Law of Syllogism, write a new conditional statement that follows from the pair of true statements.

If you study, then you will pass the test. If you pass the test, then you will pass the class.

• If you study, then you will pass the class.

• If you pass the test, then you should study.

• If you pass the class, then you will pass the test.

• If you study, then you will pass the test.

Multiple Choice

Using the law of contrapositive, write a new conditional statement given the following:

If the airport is crowded, then I will miss my flight.

I did not miss my flight.

• The airport was not crowded.

• The airport was crowded.

• Security was crowded.

• I took another flight.

Multiple Choice

If my team wins the game, then they will be the champions.

My team is the champion.

Therefore, my team won the game.

Which law is being demonstrated?

• Law of Detachment

• Law of Syllogism

• Law of Contrapositive

• No law

Multiple Choice

If I comb my hair, then I won't have tangles.

I have tangles.

Therefore, I didn't comb my hair.

Which law is being demonstrated?

• Law of Detachment

• Law of Syllogism

• Law of Contrapositive

• No law

Multiple Choice

If I see a black bear, then I will scream.

If I scream, then the bear will run away.

Therefore, if I see a black bear, it will run away.

Which law is being demonstrated?

• Law of Detachment

• Law of Syllogism

• Law of Contrapositive

• No law

Multiple Choice

If I go camping, then I need to pack my tent.

I went camping.

Therefore, I packed my tent.

Which law is being demonstrated?

• Law of Detachment

• Law of Syllogism

• Law of Contrapositive

• No law

Multiple Choice

If you need help, then you stay after class.

If you stay after class, then you ask questions.

What is the logical conclusion?

• Therefore, if you stay after class, then you need help.

• Therefore, if you ask questions, then you need help.

• Therefore, if you need help, then you ask questions.

• There is no logical conclusion.

Multiple Choice

If you go to the mall, then you will spend money.

You didn't spend any money today.

What is the logical conclusion?

• Therefore, you didn't go to the mall.

• Therefore, you don't have any money.

• Therefore, you went to the mall.

• There is no logical conclusion.

Multiple Choice

If you don't feel well, then you stay home.

You feel well.

What is the logical conclusion?

• Therefore, you stay home.

• Therefore, you don't stay home.

• Therefore, you don't feel well.

• There is no logical conclusion.

Multiple Choice

Which law is this?

a→b

b→c

∴a→c

• Law of Syllogism

• Law of Detachment

• Law of Contrapositive

• No Law

Multiple Choice

Using the Laws of Logic, what is the conclusion based on the statements given:

If a plane flies to China, then its a long flight.

Sarah bought a ticket to China.

• Sarah will get stuck in the security line.

• Sarah will have a long flight.

• Sarah will need to bring a neck pillow and headphones.

• Sarah will get stuck with a middle seat on the plane.

• No Conclusion

Multiple Choice

Which law of logic is this:

If its Sunday, then tomorrow is Monday .

Tomorrow is not Monday

Therefore, it is not Sunday

• Law of Detachment

• Law of Contrapositive

• Law of Syllogism

• No Law

Multiple Choice

Which law is this?

a→b

~b

∴~a

• Law of Syllogism

• Law of Detachment

• Law of Contrapositive

• No Law

Multiple Choice

Which law of logic is this:

If it is July, then school is out.

If school is out, then Ms. Prose is happy.

Therefore, if it is July, then Ms. Prose is happy.

• Law of Detachment

• Law of Contrapositive

• Law of Syllogism

• No Law

Multiple Choice

If $$\angle A$$  and  $$\angle A$$   are vertical angles, then    and  $$\angle B$$    are congruent.   $$\angle A$$  and $$\angle B$$   are not congruent Therefore, _________________________.

• $$\angle A$$  and  $$\angle B$$  are veritical angles.

• undefined and undefined are not veritcal angles.

• If undefined and undefined are not congruent, then undefined and undefined are not vertical angles.

• Ifundefined and undefined are not vertical angles, then undefined and undefined are not congruent.

Multiple Choice

Determine the law of logic that is presented:

If an angle measures less than 90⁰, then it is an acute angle.

The angle measures less than 90⁰.

Therefore, it is an acute angle.

• Law of Detachment

• Law of Contrapositive

• Law of Syllogism

• No Valid Law

Multiple Select

Using deductive reasoning, if the first two statements are true, then which third statements are invalid?

• $$\sim p\rightarrow\sim q$$

  $$\sim q\rightarrow\sim v$$ $$\therefore\sim p\longrightarrow\sim v$$

• $$p\rightarrow q$$
  $$q\rightarrow v$$
  $$\therefore\sim v\longrightarrow\sim p$$
  

• $$p\rightarrow q$$
  $$q\rightarrow v$$
  $$\therefore p\longrightarrow v$$
  

• $$p\rightarrow q$$ $$q\rightarrow\sim v$$ $$\therefore v\longrightarrow p$$

Multiple Choice

If $$p\rightarrow q$$ and  $$q\rightarrow b$$ , then-

• $$\sim p\rightarrow q$$  

• $$\sim p\rightarrow\sim q$$  

• $$p\rightarrow b$$  

• $$b\rightarrow p$$  

Multiple Choice

If $$\sim q\rightarrow\sim p$$  and  $$\sim p\rightarrow\sim r$$  , then-

• $$\sim q\rightarrow\sim r$$  

• $$q\rightarrow p$$  

• $$\sim p\rightarrow q$$  

• $$\sim r\rightarrow\sim q$$  

Multiple Choice

These statements given are true.

Statement: If Brett gets up early, then he will go fishing.

Statement: If Brett goes fishing, then he will catch a fish.

Using the law of syllogism, which of the following is a valid conclusion?

• If Brett does not catch a fish, then he did not go fishing.

• If Brett gets up early, then he will catch a fish.

• If Brett does not get up early, then he will not catch a fish.

• If Brett catches a fish, then he got up early.

Multiple Choice

Using deductive reasaoning, if the first two statements are true, then which third statement is valid?

• $$p\rightarrow q$$ $$q\rightarrow z$$ $$\therefore\sim z\longrightarrow\sim p$$

• $$p\rightarrow q$$
  $$q\rightarrow z$$
  $$\therefore\sim p\longrightarrow z$$
  

• $$p\rightarrow q$$
  $$q\rightarrow z$$
  $$\therefore z\longrightarrow\sim p$$
  

• $$p\rightarrow q$$
  $$q\rightarrow z$$
  $$\therefore p\longrightarrow z$$
  

Multiple Choice

Determine whether or not the argument is logically valid. If the statement is valid, state the law of logic represented. If the statement is not valid write, “No Valid Conclusion”. $$\sim R\rightarrow\sim S$$

$$\sim R$$ $$\therefore\sim S$$  

• Law of Detachment

• Law of Contrapositive

• Law of Syllogism

• No Valid Conclusion

Multiple Choice

Determine whether or not the argument is logically valid. If the statement is valid, state the law of logic represented. If the statement is not valid write, “No Valid Conclusion”. $$R\rightarrow P$$   $$T\rightarrow\sim R$$   $$\therefore R\longrightarrow T$$  

• Law of Detachment

• Law of Contrapositive
  

• Law of Syllogism

• No Valid Conclusion

Multiple Choice

Determine whether or not the argument is logically valid. If the statement is valid, state the law of logic represented. If the statement is not valid write, “No Valid Conclusion”. $$P\rightarrow Q$$   $$Q\rightarrow S$$   $$S\rightarrow T$$   $$\therefore P\longrightarrow T$$  

• Law of Detachment

• Law of Contrapositive

• Law of Syllogism

• No Valid Conclusion

Multiple Choice

Determine whether or not the argument is logically valid. If the statement is valid, state the law of logic represented. If the statement is not valid write, “No Valid Conclusion”. $$\sim P\rightarrow\sim R$$   $$R$$   $$\therefore P$$  

• Law of Detachment

• Law of Contrapositive

• Law of Syllogism

• No Valid Conclusion