Truth of the Contrapositive

Date: 06/07/2003 at 18:30:46
From: Michele
Subject: Truth tables

The inverse of a statement's converse is the statement's 
contrapositive. True, but why?

I don't know how to explain it. I tried an example:

   p: I like cats.
   q: I have cats.
   Converse  If I have cats, then I like cats.
   Inverse   If I don't like cats, then I don't have cats.
   Contrapositive  If I don't have cats, then I don't like cats.

I still can't explain the answer "true" that I came up with. Maybe it 
is wrong.

Date: 06/09/2003 at 17:03:00
From: Doctor Achilles
Subject: Re: Truth tables

Hi Michele,

Thanks for writing to Dr. Math.

The contrapositive is true if and only if the original statement is 
true. It is false if and only if the original statement is false. So 
it is logically equivalent to the original statement.

Let's say you have a conditional statement: "if I like cats, then I 
have cats." What does this mean? When is it true? When is it false?

Well, for starters, if you like cats and you have cats, then the 
conditional will come out true. That is, (P -> Q) is true when P and 
Q are both true.

Also, if you don't like cats and you don't have cats, then the 
conditional will come out true. That is, (P -> Q) is true when P and 
Q are both false.

Also, if you don't like cats and you have cats, then the conditional 
still comes out true. Remember, it says that if you like cats, then 
you will have them; it makes NO claim at all about what will happen if 
you don't like cats. So, (P -> Q) is true when P is false and Q is 
true.

However, if you like cats and you don't have cats, then the 
conditional will come out false. It says that if you like cats, then 
you will have them. So it is proven wrong if you like cats, but you 
still don't have any. So, (P -> Q) is false when P is true and Q is 
false.

So to review, (P -> Q) is true under any of these conditions:

  P is true and Q is true
  P is false and Q is true
  P is false and Q is false

And it is only false under this one condition:

  P is true and Q is false

You can say that another way, using ~P and ~Q (not-P and not-Q).

(P -> Q) is true under any of these conditions:

  ~P is false and ~Q is false
  ~P is true and ~Q is false
  ~P is true and ~Q is true

And it is only false when:

  ~P is false and ~Q is true

Is there another sentence that uses P and Q that is only false when 
~P is false and ~Q is true? Yes, the sentence is:

  (~Q -> ~P)

You can go through the same analysis of this sentence as I did for 
(P -> Q) and you'll find that it has the same truth conditions.

You can also understand this more intuitively:

The sentence:

  "If I like cats, then I have cats."

says that as long as the first part, "I like cats," is true, the 
second part, "I have cats," will definitely be true. In this case, 
what does "I don't have cats" mean? The only way "I don't have cats" 
can happen is if "I like cats" is false. That is, the only way "I 
don't have cats" can be true is if "I don't like cats" is true.  
Therefore, "If I don't have cats, then I don't like cats."

I hope this helps. If you have other questions or you'd like to talk 
about this some more, please write back.

- Doctor Achilles, The Math Forum
  http://mathforum.org/dr.math/