Contrapositive

Funny word, isn’t it? The contrapositive is the one logically valid deduction you can make from an if then statement. Let’s take the first “if….then” statement:

“Cats have tails” C –> T

I said it would be a mistake to negate or reverse these terms. But if you do both, it works. It’s true that:

“Anything without a tail is not a cat”* T –> C

If I tell you that all nerds are smart (N –>S) then it must be true that anyone who is not smart is not a nerd (N –> S)

Just reverse the terms, and put a line through them (or remove the line, if there is one).

Cars cannot float, C –> F becomes F –> C

Combining Statements Using the Contrapositive

“Elephants are heavy” “Nothing which can walk on ice is heavy”.**     E –> H,   I –> H

We can only combine from necessary to sufficient. To combine these two statements, just take the contrapositive of the second. Then we get E –> H –> I. Elephants cannot walk on ice.

Next:  How to Diagram a Statement You Don’t Understand

*This is an incredibly broad statement; many logical statements are. A car (which does not have a tail) is not a cat. Same goes for a (tail-less house). This contrapositive statement allows us to make that definite (and somewhat silly) conclusion about anything which does not have a tail.

**”No” statements sometimes confuse students. “No cat has wings”. Cat is the sufficient condition; take the “no” and use it to negate the necessary condition. C –> W.

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