The following 2 statements are logically equivalent:
- If A, then B
- If not A, then not B
Consider the following example:
There are 5 cat cards, 4 dog cards, and 1 rabbit card on the table. They belong to Tim or Sam.
We arranged these pet cards as follows, with a sign indicating what they are:
Dog, Cat, Rabbit, Sam, Tim, Cat, Tim, Tim, Sam, Dog
How do we verify if the following statements are true or false?
- All dogs are Tim's
- Only Sam has cats
Before we solve the above problem, let's first translate the above statements into a more logically normal format, and derive their logical equivalent form:
- If it is a dog (if A), then it is Tim's pet (then B)
- If it is not Tim's pet (if not B), then it is not a dog (then not A)
- If it is not Sam's pet (if not A), then it is not a cat (then not B)
- If it is a cat (if B), then it is Sam's pet (then A)
As you can see, you have 2 ways to confirm if statement #1 is right or wrong:
- You can pick a card with a "dog" sign and see if it is Tim's. If it is not, then statement #1 is proven to be wrong; if it is, then you got evidence to support statement #1 (but you have not conclusively proven statement #1 to be right)
- You can also pick a card with a "Sam" sign and see if it is a dog. If it is, then you proved statement #1 wrong; if it is not, then you got evidence to support statement #1 (but you have not conclusively proven statement #1 to be right)
You also have 2 ways to confirm if statement #2 is right or wrong:
- You can pick a card with a "Tim" sign and see if it is a cat. If it is, then you proved statement #2 wrong; if it is not, then you got evidence to support statement #2 (but you have not conclusively proven statement #2 to be right)
- You can pick a card with a "cat" sign and see if it is Sam's. If it is not, then you proved statement #2 wrong; if it is, then you got evidence to support statement #2 (but you have not conclusively proven statement #2 to be right)
Think for yourself how to confirm the following statements:
- At least 1 dog is Sam's
- Some cats are Tim's
- Tim only has rabbits
- All rabbits are Sam's pets