Contraposition

To obtain the contrapositive of a categorical proposition (1) switch the subject and predicate, and (2) replace the subject and predicate with their complements

First, consider A-statements

All S are P 
Contrapose
All non-P are non-S

Let’s use an example to help us determine whether contraposing an A statement is valid.

All dogs are mammals
Contrapose
All non-mammals are non-dogs

The original states that all members of the dog class are members of the mammal class. The dog class is completely contained within the mammal class. 
The contrapositive states that all members of the non-mammal class are members of the non-dog class. The non-mammal class is completely contained within the non-dog class.
            
In other words, if x is a non-mammal, then x is a non-dog

Intuitively, it does seem like both the original A and contrapositive A are saying the same thing. If the original A is true, then contrapositive A must be true as well. 

Let’s no consider E-statements

No S are P
Contrapose
No non-P are non-S

Let’s use an example

No dogs are cats
Contrapose
No non-cats are non-dogs

The original states that no member of the dog class is a member of the cat class. The two classes are completely separated. According to the original, these classes do not overlap at all. 
            
In other words, if x is a dog, then x is not a cat

The contrapositive states that no member of the non-cat class is a member of the non-dog class. According to the contrapositive, these classes do not overlap at all. 
            
In other words, if x is a non-cat, then x is not a non-dog

Intuitively, it seems that it is possible for the original E to be true and the contrapositive E to be false. 

It is true that no cats are dogs. It is false that no non-cats are non-dogs. I am a non-cat and a non-dog. The computer you are looking at is a non-cat and a non-dog.

Let’s dig a bit deeper into the E and its contrapositive. 

When we say that No S are P we are saying that the S class and the P class are completely separate—they have no common members. Nothing in the S class is in the P class (and of course the converse is true as well—nothing in the P class is in the S class).

When we say that No non-P are non-S we are saying that the non-P class and the non-class are completely separate—they have no common members. Nothing in the non-P class is in the non-S class (and of course the converse is true as well—nothing in the non-S class is in the non-P class)

In other words, we are saying that there are no members of both the non-P class and the non-S class. So, the area that represents both the non-P class AND the non-S class is empty. So, we have to locate that area. We have to find the area that represents both the non-P class and the non-S class and make it empty.

            How can we say this in terms of S and P? 

            Where is the non-P class AND the non-S class? 

                        The area outside the S class is the non-S class.
                        The area outside the P class is the non-P class.
  So, the area outside both the S class and the P class is the area of the non-S  
  AND non-P class. 
  So, that area is empty. 
            
            


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