More on Logic

It all began with babies and crocodiles, courtesy of Lewis Carroll.

1. Babies are illogical;
2. Nobody is despised who can manage a crocodile;
3. Illogical persons are despised.

Each of these statements can be written as an implication, a statement of the form "A implies B" or "if A, then B". As mathematicians, we abbreviate such a statement by A -> B. We can replace the statements about babies and crocodiles with implications.

1. baby -> illogical
2. manage crocodile -> not despised
3. illogical -> despised

From the statement A -> B we can conclude that its contrapositive (not A) -> (not B) must also be true, so we can replace our implication for statement 2 with

despised -> cannot manage crocodile.

Then we string the chains of implications together:

baby -> illogical -> despised -> cannot manage crocodile.

Turning this back into English, we conclude that babies cannot manage crocodiles.

After doing several of these puzzles, we had the kids come up with their own. Here are some of their puzzles:

1. All doughnuts are fattening.
2. 0 calorie food is not fattening.
3. Paczkis are big doughnuts.
4. Meijer is selling 0 calorie Paczkis. Can this be true?

1. If you're cool you are a dude.
2. Luke is not a dude.
3. If you're not cool you can't get into the club.

1. All happy people go to Summers-Knoll.
2. All people at Summers-Knoll are smart.
3. Victor does not go to Summers-Knoll.

In the last puzzle, we can conclude that Victor does not go to Summers-Knoll. However, we can't conclude that Victor is not smart. From

SK -> smart

we can conclude the contrapositive:

not smart -> not SK

but we can NOT conclude

not SK -> not smart.

There could be smart people who don't go to Summers Knoll!