Counterexamples and Invalidity
Fallacy of denying the antecedent
Similar to modus tollens, but is a fallacy.
An invalid argument form; If A, then B; not A; so, not B.
Invalid Argument Form
One that has some invalid substitution instances.
If Britney Spears is an oil-tycoon, then Britney Spears is rich.
Britney Spears isn’t an oil-tycoon.
So, Britney Spears isn’t rich.
Fallacy of denying the antecedent
Similar to modus tollens, but is a fallacy.
An invalid argument form; If A, then B; not A; so, not B.
Invalid Argument Form
One that has some invalid substitution instances.
If Britney Spears is an oil-tycoon, then Britney Spears is rich.
Britney Spears isn’t an oil-tycoon.
So, Britney Spears isn’t rich.
Counterexample
A counterexample to an argument form is a substitution instance in which the premises are true and the conclusion is false.
If there are Beefsteaks in Dan’s summer garden, then there are tomatoes in it.
There are not Beefsteaks in Dan’s summer garden.
So, there are no tomatoes in Dan’s summer garden.
Good Counterexample
A good counterexample to an argument form in a substitution instance in which the premises are well-known truths and the conclusion is a well know falsehood.
Britney Spears Example
Fallacy of Affirming the Consequent
Similar to modus ponens, but is a fallacy.
An invalid argument form; If A, then B; B; So, A.
Counterexample Method
Argument from someone who doesn’t understand logic:
George Bush was a U.S. president and George Bush went to Harvard.
Barack Obama is a U.S. president.
So, Barack Obama went to Harvard.
Form
A and B
C
So, D
Counterexample from someone who understands logic:
George Washington was a U.S. president and George Washington is long dead.
Barack Obama is a U.S. president
So, Barack Obama is long dead.
Or…
John Lennon was a member of the Beatles and Fuji apples are red.
Not everyone like escargots de Bourgogne.
So, the Moon is made of limburger.
Counterexample Method cont.
Bad argument…
If stem-cell research causes harm, then stem-cell research is wrong.
If stem-cell research causes harm, then it should be outlawed.
So, If stem-cell research is wrong, it should be outlawed
Form
If A, then B.
If A, then C.
So, if B, then C.
Counterexample
If Johnny Depp is a thoroughbred, then he is a mammal.
If Johnny Depp is a thoroughbred, then he is a horse.
So, if Johnny Depp is a mammal, then he is a horse.
Categorical Statement
A statement that relates two classes or categories, where a class is a set or collection of things.
All presidents are human beings.
All human beings are mammals.
So, all presidents are mammals.
Term
A word or phrase that stands for a class of things.
Form
All A are B.
All B are C.
So, all A are C.
Think of a Venn diagram.
Categorical Statement Example
All emeralds are gems.
Some rocks are not gems.
So, some rocks are not emeralds.
Form
All A are B.
Some C are not B.
So, some C are not A.
Sounds stupid but is valid.
Some = at least 1. So, ‘some rocks are not emerald' means ‘at least 1 rock is not an emerald.’
A counterexample to an argument form is a substitution instance in which the premises are true and the conclusion is false.
If there are Beefsteaks in Dan’s summer garden, then there are tomatoes in it.
There are not Beefsteaks in Dan’s summer garden.
So, there are no tomatoes in Dan’s summer garden.
Good Counterexample
A good counterexample to an argument form in a substitution instance in which the premises are well-known truths and the conclusion is a well know falsehood.
Britney Spears Example
Fallacy of Affirming the Consequent
Similar to modus ponens, but is a fallacy.
An invalid argument form; If A, then B; B; So, A.
Counterexample Method
Argument from someone who doesn’t understand logic:
George Bush was a U.S. president and George Bush went to Harvard.
Barack Obama is a U.S. president.
So, Barack Obama went to Harvard.
Form
A and B
C
So, D
Counterexample from someone who understands logic:
George Washington was a U.S. president and George Washington is long dead.
Barack Obama is a U.S. president
So, Barack Obama is long dead.
Or…
John Lennon was a member of the Beatles and Fuji apples are red.
Not everyone like escargots de Bourgogne.
So, the Moon is made of limburger.
Counterexample Method cont.
Bad argument…
If stem-cell research causes harm, then stem-cell research is wrong.
If stem-cell research causes harm, then it should be outlawed.
So, If stem-cell research is wrong, it should be outlawed
Form
If A, then B.
If A, then C.
So, if B, then C.
Counterexample
If Johnny Depp is a thoroughbred, then he is a mammal.
If Johnny Depp is a thoroughbred, then he is a horse.
So, if Johnny Depp is a mammal, then he is a horse.
Categorical Statement
A statement that relates two classes or categories, where a class is a set or collection of things.
All presidents are human beings.
All human beings are mammals.
So, all presidents are mammals.
Term
A word or phrase that stands for a class of things.
Form
All A are B.
All B are C.
So, all A are C.
Think of a Venn diagram.
Categorical Statement Example
All emeralds are gems.
Some rocks are not gems.
So, some rocks are not emeralds.
Form
All A are B.
Some C are not B.
So, some C are not A.
Sounds stupid but is valid.
Some = at least 1. So, ‘some rocks are not emerald' means ‘at least 1 rock is not an emerald.’