Exercises §1.1 #9(a) and §1.2 #7

Exercise §1.2 #7 is an unusually creative homework problem, and I like it more the more I think about it. It asks you to return to the argument from §1.1 #9(a) and check your earlier intuition using the new tool from §1.2: a truth table.

The argument

Let

JM mean “Jane wins the math prize,”
PM mean “Pete wins the math prize,” and
PC mean “Pete wins the chemistry prize.”

We do not need a proposition JC, because the statement “Jane wins the chemistry prize” never appears. The three premises and the conclusion are

NOT (JM AND PM)
PM OR PC
JM
Therefore, PC.

Since JM is true and JM AND PM is not true, PM must be false. Since PM OR PC is true and PM is false, PC must be true. This is what allows us to conclude that Pete wins the chemistry prize.

The truth-table check

For §1.2 #7, make the full truth table. The last three columns are the three premises.

JM  PM  PC | JM AND PM | NOT (JM AND PM) | PM OR PC | JM
-----------+-----------+-----------------+----------+---
 T   T   T |     T     |        F        |    T     | T
 T   T   F |     T     |        F        |    T     | T
 T   F   T |     F     |        T        |    T     | T  <-- remains
 T   F   F |     F     |        T        |    F     | T
 F   T   T |     F     |        T        |    T     | F
 F   T   F |     F     |        T        |    T     | F
 F   F   T |     F     |        T        |    T     | F
 F   F   F |     F     |        T        |    F     | F

First look at the final JM column. Since that premise must be true, eliminate the bottom four rows. Next look at PM OR PC. It eliminates the row T, F, F. Finally, NOT (JM AND PM) eliminates the two rows that begin T, T. Only the row T, F, T remains. In that row PC is true, so Pete wins the chemistry prize.