Mr. Nance,

I am having a hard time with problem 3 in Exercise 3 of Intermediate Logic.  For the first proposition, ~(J ⋅ R), the answer key says “Joe and Rachel are not both students.” For the second proposition, ~J ⋅ ~R, the answer key says “Both Joe and Rachel are not students.” Those sound the same to me.

The difference is this. To say “Joe and Rachel are not both students” still means that one of them could be a student. But to say “Both Joe and Rachel are not students” means that neither of them is a student. 

(That’s the simple answer; the rest of my response below is for those who want further understanding.)

That is why in problem 2, where these two propositions are symbolized, the truth tables are different. In fact, they are different when one of the simple propositions is true and the other false. If Joe is a student but Rachel is not a student (the second row), then “Joe and Rachel are not both students” is true, but “Both Joe and Rachel are not students” is false. Similarly, if Joe is not a student but Rachel is a student (the third row),  then “Joe and Rachel are not both students” is true, but “Both Joe and Rachel are not students” is false.

The truth tables have the same value for the first and fourth rows. In the first row (T T), Joe is a student and Rachel is a student. In that case, “Joe and Rachel are not both students” is false (because both of them are students). Also, “Both Joe and Rachel are not students” is clearly false. 

In the fourth row (F  F), Joe is not a student and Rachel is not a student. In this situation, “Joe and Rachel are not both students” is true. But more clearly, “Both Joe and Rachel are not students” is true.

It might help you to picture actual Joes and Rachels, students and non-students, that you personally know as you work through this.

Blessings!