If you took a true if-then statement and inserted a not in each clause and reversed the clauses would the new statement also be true?

Answer 1

Ordinarily yes. This is called "contraposition" in formal logic :

For any statement where A implies B, then not B always implies not A. Proving or disproving either one of these statements automatically proves or disproves the other.

If A then B means that B is a subset of A, and anything not part of B cannot be part of A.

However, some if-then or cause-effect statements, once made negative or reversed, represent an entirely different fact that may or may not be true.

For example:

"If I get the high score, then I will receive the award."

would be true in contraposition if the award is based only on the score.

"If I do not receive the award, then I did not get the high score."

However, if the award was denied on some other basis, the contraposition is not valid.

The valid statement would be "If I get the high score, then I will definitely get the award."

Answer 2

Not necessarily. The reverse antecedent does not guarantee the reverse condition. This is the logical fallacy called "denying the antecedent" : if A then B. Not A, not B. The status of B is not necessarily determined by A.

Examples:

If I bought bread, I went to the bakery. If I did not buy bread, I still could have gone to the bakery.

It does work for truisms, where there is only one logical possibility:

If today is Thursday, then tomorrow is Friday.

Answer 3

Ordinarily, no. Just because one condition implies another (e.g. If an animal is a bat, then it is a mammal) does not imply the other (if an animal is a mammal, then it is a bat).