Chapter B.2, Problem 7E

Solution

If it follows logically that if it does not rain, then Iris does not go to the movies.

It has been determined that it does not logically follow that if it does not rain, then Iris does not go to the movies.

Given:

Iris makes the true statement, “If it rains, then I am going to the movies.”

Concept used:

The conditional statement “If $p$ , then $q$ ” has converse of the form “If $q$ , then $p$ ”, inverse of the form “If not $p$ , then not $q$ ” and contrapositive of the form “If not $q$ , then not $p$ ”.

An implication and its contrapositive are logically equivalent. The converse and the inverse of a conditional statement are logically equivalent.

Calculation:

It is given that Iris makes the true statement, “If it rains, then I am going to the movies.”

The inverse of the above conditional statement is:

“If it does not rain, then I am not going to the movies.”

However, an implication and its inverse are not logically equivalent.

So, the truth values for an implication and its inverse may not be the same.

This implies that while it is true that if it rains, then Iris goes to the movies, it is not necessarily true that if it does not rain, then Iris does not go to the movies.

Thus, it does not logically follow that if it does not rain, then Iris does not go to the movies.

Conclusion:

It has been determined that it does not logically follow that if it does not rain, then Iris does not go to the movies.