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"If A then B" vs "If not B, then not A"

The following 2 statements are logically equivalent:

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?

  1. All dogs are Tim's
  2. 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: 

  1. If it is a dog (if A), then it is Tim's pet (then B)
    1. If it is not Tim's pet (if not B), then it is not a dog (then not A)
  2. If it is not Sam's pet (if not A), then it is not a cat (then not B)
    1. 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:

  1. 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)
  2. 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:

  1. 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)
  2. 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:

  1. At least 1 dog is Sam's
  2. Some cats are Tim's
  3. Tim only has rabbits
  4. All rabbits are Sam's pets